Nonstationary Gabor Frames - Approximately Dual Frames and Reconstruction Errors

0.49 MB·English

by  Monika Dörfler

Nonstationary Gabor Frames - Approximately Dual
Frames and Reconstruction Errors
3
1
Monika D¨orfler, Ewa Matusiak
0 ∗†
2

n January 10, 2013
a
J

9

] Abstract
A
F Nonstationary Gabor frames, recently introduced in adaptive signal anal-
.
h ysis, represent a natural generalization of classical Gabor frames by allowing
t
a
for adaptivity of windows and lattice in either time or frequency. Due to the
m
lack of a complete lattice structure, perfect reconstruction is in general not
[

1 feasible from coefficients obtained from nonstationary Gabor frames. In this
v
paper it is shown that for nonstationary Gabor frames that are related to some
2
0 known frames for which dual frames can be computed, good approximate re-
8
1 construction can be achieved by resorting to approximately dual frames. In
.
1 particular, we give constructive examples for so-called almost painless nonsta-
0
3 tionary frames, that is, frames that are closely related to nonstationary frames
1
: with compactly supported windows. The theoretical results are illustrated by
v
i concrete computational and numerical examples.
X
r
a
Keywords: adaptive representations, nonorthogonal expansions, irregular
Gabor frames, reconstruction, approximately dual frame
1 Introduction
Adapted and adaptive signal representation have received increasing interest over
the past few years. As opposed to classical approaches such as the short-time Fourier
∗This work was supported by the WWTF project Audiominer (MA09-24)
†The authors are with the Department of Mathematics, NuHAG, University of Vi-
enna, Nordbergstrasse 15, 1090 Wien, Austria (e-mail: monika.doerfl[email protected],
[email protected])
1
transform(STFT)orwavelet transform, adaptiverepresentations allowforavariation
ofparameters such aswindow width orsampling density over time, frequency orboth.
Changing parameters in the frequency domain leads, for example, to non-uniform
filter banks while adapting window width and sampling density in time is reminis-
cent of the approach suggested in the construction of nonuniform lapped transforms.
Transforms featuring simultaneous adaptivity in time and frequency are notoriously
difficult to construct and implement, cp. [15, 7, 12]; however they have shown to be
useful in some applications , cf. [14]. On the other hand, fast and efficient imple-
mentations exist for representations with adaptivity in only time or frequency. One
recent method to obtain this kind of representations is represented by nonstationary
Gabor frames, first suggested in [11] and further developed in [1, 18, 10]. All the
known implementations rely on compactness of the used analysis window in either
time of frequency. This assumption allows for usage of tools developed for pain-
less non-orthogonal expansions [6]. While a priori very convenient, the restriction to
using compactly supported windows in the domain for which one wishes a flexible
representation can be undesirable. As an example, we mention the construction of
nonuniform filter banks via nonstationary Gaborframes, in which case this restriction
forbids finite impulse response (FIR) filters; the latter are, however, imperative for
real-time processing applications.
In the current contribution, we therefore go beyond the results presented in the
references above and consider nonstationary Gabor frames with fast decay but un-
bounded support. The existence of this kind of frames was shown in [8]. Here we are
concerned with methods for approximate reconstruction for these adaptive systems.
2 Notation and Preliminaries
Given a non-zero function g L2(R), a modulation, or frequency shift, operator M
bl
∈
is defined by M g(t) := e2πibltg(t), and time shift operator T by T g(t) := g(t ak).
bl ak ak
−
A composition, g = M T g(t) := e2πibltg(t ak) is a time-frequency shift operator.
k,l bl ak
−
The set (g,a,b) = g : k,l Z is called a Gabor system for any real, positive
k,l
G { ∈ }
a,b. (g,a,b)isaGaborframeforL2(R), ifthereexist framebounds0 < A B <
G ≤ ∞
such that for every f L2(R) we have
∈
A f 2 f,g 2 B f 2. (1)
k k2 ≤ |h k,li| ≤ k k2
k,l Z
X∈
2
When working with irregular grids, we assume that the sampling points form
a separated set: a set of sampling points a : k Z is called δ-separated, if
k
{ ∈ }
a a > δ for a , a , whenever k = m. χ will denote the characteristic function
k m k m I
| − | 6
of the interval I.
A convenient class of window functions for time-frequency analysis on L2(R) is
the Wiener space.
Definition 1. A function g L (R) belongs to the Wiener space W(L ,ℓ1) if
∞ ∞
∈
kgkW(L∞,ℓ1) := ess supt∈Q|g(t+k)| < ∞, Q = [0,1].
k Z
X∈
For g W(L ,ℓ1) and δ > 0 we have [9]
∞
∈
esssupt∈R |g(t−δk)| ≤ (1+δ−1)kgkW(L∞,ℓ1). (2)
k Z
X∈
For f L2(R) we use the following normalization of the Fourier transform, which
∈
we denote by :
F
f(ω) = f(ω) = f(t)e 2πiωtdt.
−
F
R
Z
b
3 Nonstationary Gabor Frames
Nonstationary Gabor systems are a generalization of classical Gabor systems of reg-
ular time-frequency shifts of a single window function.
Definition 2. Let g = g W(L ,ℓ1) : k Z be a set of window functions and
k ∞
{ ∈ ∈ }
let b = b : k Z be a corresponding sequence of frequency-shift parameters. Set
k
{ ∈ }
g = M g . Then, the set
k,l bkl k
(g,b) = g : k,l Z (3)
k,l
G { ∈ }
is called a nonstationary Gabor (NSG) system.
In generalization of regular Gabor frames, for which g = T g, we will usually
k ak
assume that the windows g are localized around points a in a separated set of time-
k k
sampling points a : k Z . Further, we will always make the assumption that the
k
{ ∈ }
frequency sampling parameters b are positive numbers contained in a closed interval,
k
i.e. b [b ,b ] R+ for all k Z.
k L U
∈ ⊂ ∈
3
To every collection (3) we associate the analysis operator C given by (C f) =
g g k,l
f,g , and synthesis operator U , where U c = c g and c ℓ2. For two
h k,li g g k,l Z k,l k,l ∈
∈
Gabor systems (g,b) and (γ,b) the composition S = U C ,
P g,γ γ g
G G
S f = f,M g M γ , (4)
g,γ h lbk ki lbk k
k,l Z
X∈
admits a Walnut representation for all f L2(R), [8]:
∈
S f(t) = b 1g (t lb 1)γ (t)f(t lb 1). (5)
g,γ −k k − −k k − −k
k,l Z
X∈
We will frequently use the following correlation functions of a pair of Gabor systems:
Gg,γ(t) = b 1 g (t lb 1) γ (t) , for l Z. (6)
l −k | k − −k || k | ∈
k Z
X∈
Note that this definition is asymmetric with respect to g and γ. Using this notation,
we obtain the following bounds for the frame operator (4):
S 2 esssup Gg,γ esssup Gγ,g. (7)
k g,γk ≤ l · l
l Z l Z
X∈ X∈
By inspection of (5), we note that the summands corresponding to l = 0 may be seen
6
as the off-diagonal entries of the frame operator. We thus isolate the diagonal part
Gg,γ(t) = b 1 g (t) γ (t) (8)
0 −k | k || k |
k Z
X∈
and denote the off-diagonal entries as follows:
R = esssup b 1 g ( lb 1) γ ( ) . (9)
g,γ −k | k ·− −k || k · |
l Z 0 k Z
∈X\{ }X∈
Note that, if g = γ, then the diagonal part of the frame operator S is equal to Gg,g.
g,g 0
Using this notation, we obtain the following additional bound:
S f,f f 2 esssupGg,γ + R R , (10)
h g,γ ik k−2 ≤ 0 g,γ · γ,g
p
Bessel sequences are of particular importance in the theory of frames and Riesz
bases,[5, 19]. In the regular Gabor case, where g (t) = g(t ak) for some a > 0, it is
k
−
sufficient to assume g W(L ,ℓ1) to obtain a Bessel sequence. We next provide a
∞
∈
generalization of this property to NSG frames.
4
Proposition 3.1. Let (g,b) be a NSG system. If g W(L ,ℓ1) for all k Z
k ∞
G ∈ ∈
with supk∈ZkgkkW(L∞,ℓ1) bounded, and k∈Z|gk(t)| ≤ B almost everywhere for some
B < , then the sequence g is a Bessel sequence.
k,l P
∞
Proof. Let f L2(R). Then by the assumption on the windows g and estimate (7)
k
∈
f,g 2 = S f,f f 2 esssup Gg,g
|h k,li| h g,g i ≤ k k2 · l
k,l Z l Z
X∈ X∈
= f 2 esssup b 1 g ( ) g ( lb 1)
k k2 · −k | k · | | k ·− −k |
k Z l Z
X∈ X∈
≤ kfk22 ·esssup b−k1|gk(·)|(1+b−k1)kgkkW(L∞,ℓ1)
k Z
X∈
≤ kfk22 ·B ·skupZ[(1+bk)kgkkW(L∞,ℓ1)].
∈
Given a frame, it is well known that there exists at least one dual frame (γ,b)
G
such that
f = f,γ g , for all f L2(R). (11)
k,l k,l
h i ∈
k,l Z
X∈
The canonical dual frame is given by γ = S 1g . In the regular Gabor case,
k,l − k,l
the dual frames are again Gabor frames, i.e., they consist of time-frequency shifted
versions of one dual window. This is due to the fact, that the frame operator S com-
muteswithtime-frequencyshifts, henceγ = S 1g = S 1M T g = M T S 1g =
k,l − k,l − bl ak bl ak −
M T γ. In general, we cannot expect, that the dual frame of a NSG frame is again
bl ak
a NSG frame.
However, even in the case of regular Gabor frames, it is often difficult to calculate a
dual frame explicitly. For that reason, alternative possibilities to obtain perfect or ap-
proximate reconstruction have been proposed, [4, 2]. The following lemma quantifies
the reconstruction error using general pairs of Bessel sequences.
Lemma 3.2. Let (g,b) and (γ,b) be two Bessel sequences. Then
G G
I S 1 b 1g γ + R R . (12)
k − g,γk ≤ − −k k k g,γ · γ,g
(cid:13) Xk∈Z (cid:13)∞ p
(cid:13) (cid:13)
(cid:13) (cid:13)
Proof. Starting from the Walnut representation of S , we estimate using Cauchy-
g,γ
Schwartz inequality for sums and integrals and, since all summands have absolute
value, Fubini’s theorem to justify changing the order of summation and integral:
5
f S f,f = f b 1g γ f,f b 1γ ( )g ( lb 1)f( lb 1),f
|h − g,γ i| h − −k k k i− −k k · k ·− −k ·− −k
(cid:12)(cid:12) Xk∈Z Dl∈XZ\{0}Xk∈Z E(cid:12)(cid:12)
(cid:12)1 b 1g γ f 2 + R R f 2, (13) (cid:12)
≤ − −k k k k k2 g,γ · γ,gk k2
(cid:13) Xk∈Z (cid:13)∞ p
(cid:13) (cid:13)
since (cid:13) (cid:13)
b 1γ ( )g ( lb 1)f( lb 1),f (14)
−k k · k ·− −k ·− −k
(cid:12)(cid:12)Dl XkZ∈Z0 E(cid:12)(cid:12)
(cid:12) ∈ \{ } (cid:12)
b 1 γ (t) g (t lb 1) f(t lb 1) f(t) dt
≤ −k | k || k − −k || − −k || |
R
k Z Z
l XZ∈ 0
∈ \{ }
1/2 1/2
b 1 g (t lb 1) γ (t) f(t lb 1) 2dt g (t lb 1) γ (t) f(t) 2dt
≤ −k | k − −k || k || − −k | | k − −k || k || |
R R
k Z (cid:20)Z (cid:21) (cid:20)Z (cid:21)
l XZ∈ 0
∈ \{ }
1/2 1/2
≤  |f(t)|2 b−k1|gk(t)||γk(t−lb−k1)|dt  |f(t)|2 b−k1|γk(t)||gk(t−lb−k1)|dt .
R R
Z k Z Z k Z
 l XZ∈ 0   l XZ∈ 0 
 ∈ \{ }   ∈ \{ } 
   
A special class of NSG systems are collections of compactly supported windows.
They were first addressed in [1]. The collection (g,b) with windows g being com-
k
G
pactly supported with suppg 1 for all k is a frame for L2(R) if there exist
| k| ≤ bk
constants A > 0 and B < such that
∞
A Gg,g(t) B a.e.. (15)
≤ 0 ≤
In this situation, (g,b) is called painless NSG frame. The canonical dual atoms
G
are given by γ = M (Gg,gg) 1g . Note again that, in general, we may have γ =
k,l lbk 0 − k k,l
S 1(M g ) = M (S 1g ). If b = b for all k, then the frame operator commutes
− lbk k 6 lbk − k k
with the frequency-shifts and the dual frame is an NSG frame.
The existence of NSG frames with not necessarily compactly supported windows
was established in [8]. For these frames, finding canonical dual frames requires the in-
version of the frame operator. This computation is expensive since the operator has
considerably less structure than the frame operator in the classical, regular Gabor
frame case, for which fast algorithms now exist, [17, 13, 16]. To circumvent the prob-
lem, we suggest the use of windows other than canonical duals to obtain sufficiently
good approximate reconstruction.
6
4 Approximately dual atoms
Thenotionofapproximatelydualpairswasdiscussedin[4]. ForNSGBesselsequences
we adapt their definition as follows.
Definition 3. Two Bessel sequences (g,b)and (γ,b)aresaidtobeapproximately
G G
dual frames if I S < 1 or I S < 1.
g,γ γ,g
k − k k − k
Note that the two conditions given in the definition are equivalent since
I S = I C U = I U C = I C U = I S .
k − g,γk k − g γk k − γ∗ g∗k k − γ gk k − γ,gk
In Definition 3 it is implicitly stated that, if two Bessel sequences are approximately
dual frames, then each of them is a frame. This result was proved in [4] for general
frames and it will be useful to reformulate the conditions for the existence of NSG
frame, given in [8], in the context of approximately dual frames.
Proposition 4.1. Let (g,b) be a Bessel sequence with Bessel bound B and 0 <
G
A g (t) 2 A < a.e. for some positive constants A ,A .
1 ≤ k Z| k | ≤ 2 ∞ 1 2
∈
i) TPhe multiplication operator Gg,g is invertible a.e. and, for γ = (Gg,g) 1g ,
0 k 0 − k
R
g,g
I S . (16)
k − g,γk ≤ essinfGg,g
0
ii) If
R < essinfGg,g, (17)
g,g 0
then (g,b) and (γ,b) are approximately dual frames for L2(R).
G G
iii) Assume, additionally, for some δ-separated set of time-sampling points a :
k
{
k Z and constants 0 < p ,C ,C < such that for p ]2,p ] R,
U L U k U
∈ } ∞ ∈ ⊂
C [C ,C ] we have
k L U
∈
g (t) C (1+ t a ) pk for all k Z. (18)
k k k −
| | ≤ | − | ∈
Then there exists a sequence {b0k}k∈Z, such that for all sequence bk ≤ b0k, k ∈ Z,
(17) holds.
Remark 1. If (17) holds, (γ,b) is called a single preconditioning dual system for
G
(g,b).
G
7
Proof. Since all frequency modulation parameters b are taken from a closed interval
k
in R+, the invertibility of Gg,g is straightforward and the windows γ are well de-
0 k
fined. Moreover, since (g,b) is a Bessel sequence, so is (γ,b), with Bessel bound
G G
(essinf Gg,g) 2B. Substituting γ = (Gg,g) 1g for γ in the proof of Lemma 3.2, the
0 − k 0 − k k
first term in (13) vanishes and we obtain (i):
f S f,f (Gg,g) 1 b 1g ( )g ( lb 1)f( lb 1),f
|h − g,γ i| ≤ 0 − −k k · k ·− −k ·− −k
(cid:12)(cid:12)D l∈XZ\{0}Xk∈Z E(cid:12)(cid:12)
(cid:12)(essinfGg,g) 1 b 1 g ( ) g ( lb 1) f( (cid:12)lb 1) , f
≤ 0 − −k | k · || k ·− −k || ·− −k | | |
Dl∈XZ\{0}Xk∈Z E
(essinfGg,g) 1R f 2. (19)
≤ 0 − g,gk k2
(ii) follows directly from Definition 3. Finally, (iii) follows from [8, Theorem 3.4],
whereitisshownthattheassumptions(18)onthewindowsg guaranteetheexistence
k
of a sequence b0 such that
k
S f,f f 2 essinfGg,g R > 0. (20)
h g,g ik k−2 ≥ 0 − g,g
Single preconditioning dual windows are a good choice for reconstruction, when-
ever the frame operator is close to diagonal. This is the case, if the original windows
g decay fast and frequency sampling is fast.
k
If the frame of interest is close to some other frame, which, ideally, is better under-
stood, other approximate dual windows may be derived from this frame. The proto-
typical situation is a NSG frame which is close to a painless NSG frame in the sense of
a small perturbation. Approximately dual frames in the context of perturbation the-
ory were recently studied in [4]. In the following proposition we give error estimates
for the reconstruction with approximately dual frames in such a situation. This pro-
vides different reconstruction methods apart from single preconditioning which was
addressed in Proposition 4.1.
Proposition 4.2. Assume that (g,b) is a Bessel sequence with bound B and that
G
(h,b) is a NSG frame with lower and upper frame bound A and B , respectively.
h h
G
We set ψ = h g and define the following windows:
k k k
−
(a)γ1 =S 1h (canonical dual of h ) (21)
k,l h−,h k,l k,l
(b)γ2 =S 1g (22)
k,l h−,h k,l
Then the following hold:
8
(i)
1/2
kI −Sg,γ1k ≤ A−h kCψk. (23)
If esssup Gψ,ψ < A , then (γ1,b) and (g,b) are approximately dual
l Z l h G G
∈
frames.
P
(ii)
I S A 1( B +√B) C . (24)
k − g,γ2k ≤ −h h k ψk
If esssup Gψ,ψ < A2h , then p(γ2,b) and (g,b) are approximately
l∈Z l (√Bh+√B)2 G G
dual frames.
P
Remark 2. The first statement of Proposition 4.2 is contained in [4].
According to [8], the assumption that esssup Gψ,ψ < A can be satisfied if
l Z l h
∈
the functions ψ decay polynomially, i.e., ψ (t) C (1 + t ) pk with appropriate
k k P k −
| | ≤ | |
constants C and decay rates p > 1 .
k k
Proof. From (7) it follows that C 2 esssup Gψ,ψ. The same estimate holds
k ψk ≤ l Z l
∈
for U 2.
ψ P
k k
Since (h,b) is a frame with canonical dual frame (γ1,b), an upper frame bound
G G
of (γ1,b) is given by A 1 . We thus obtain (i) as follows:
G −h
1/2
kI −Sg,γ1k = kUγ1Ch −Uγ1Cgk ≤ kUγ1kkCψk ≤ A−h kCψk. (25)
If esssup Gψ,ψ < A , then I S < 1 and (γ1,b) and (g,b) are approx-
l Z l h k − g,γ1k G G
∈
imately dual frames as claimed.
P
To show (ii), we note that U = S 1U and thus
γ2 h−,h g
I S = S 1S S 1S = S 1(U C U C ) (26)
k − g,γ2k k h−,h h,h − h−,h g,gk k h−,h h h − g g k
S 1 U C U C +U C U C = A 1 U C U C (27)
≤k h−,hkk h h − g h g h − g gk −h k ψ h − g ψk
A 1 C ( B +√B) (28)
≤ −h k ψk h
p
and (24) follows.
4.1 Perturbation of painless nonstationary Gabor frames
IfaNSGsystemcanbederivedasaperturbationofapainlessNSGframe, theapprox-
imately dual windows given in Proposition 4.2 are particularly simple to compute. In
this situation, the frame (h,b) is the painless frame (go,b) and the frame operator
G G
9
Sh,h = Sgo,go is the multiplication operator Gg0o,go. Moreover, in this particular case,
the approximately dual frames, given by γ1 = (Ggo,go) 1go and γ2 = (Ggo,go) 1g
k,l 0 − k,l k,l 0 − k,l
are NSG frames. This is a very important asset, since for NSG frames fast algorithms
for analysis and reconstruction using FFT exist.
In [8] we constructed a special class of NSG frames, arising from painless NSG
frames, which we introduce next.
Definition 4 (Almost painless NSG frames). Let (g,b) be a NSG system, assume
G
that the windows g are essentially bounded away from zero on the intervals I =
k k
[a (2b ) 1,a + (2b ) 1] and set go = g χ . If (go,b) is a (painless) frame for
k − k − k k − k k Ik G
L2(R), then we call the system (g,b) an almost painless NSG system (or frame).
G
For almost painless NSG systems, the estimates given in Proposition 4.2 can be
written more explicitly.
Corollary 4.3. Assume that (g,b) is an almost painless NSG system and let A =
0
G
essinf Ggo,go to be the lower frame bound of the painless frame (go,b) and gr =
0 G k
g g χ . Then the following hold:
k − k Ik
(i) for γ1 = (Ggo,go) 1go,
k 0 − k
kI −Sg,γ1k ≤ A−01 Rgr,go ·Rgo,gr (29)
p
(ii) for γ2 = (Ggo,go) 1g
k 0 − k
kI −Sg,γ2k ≤ A−01 Rgr,go +Rgo,gr +esssup Gglr,gr (30)
!
l Z
X∈
Proof. The estimates follow from Lemma 3.2. First, substituting γ1 = (Ggo,go) 1go
k 0 − k
for γ in (12), the first term vanishes, since b 1g go = Ggo,go, and we obtain
k k Z −k k k 0
∈
kI −Sg,γ1k ≤ Rg,γ1 ·Rγ1,g ≤ (esPsinfGg0o,go)−1 Rgr,go ·Rgo,gr ,
p p
since
R = esssup b 1 g ( lb 1) (Ggo,go) 1go( ) (31)
g,γ1 −k | k ·− −k || 0 − k · |
l Z 0 k Z
∈X\{ }X∈
(essinfGgo,go) 1 esssup b 1 gr( lb 1) go( )
≤ 0 − · −k | k ·− −k || k · |
l Z 0 k Z
∈X\{ }X∈
= (essinfGg0o,go)−1Rgr,go,
10