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Parametric Wavelets: Tight Frames Class
Copyright © IFAC System Structure and Control, Bucharest, Romania, 1997
PARAMETRIC WAVELETS: TIGHT FRAMES CLASS
DaIl $teIaIloiu "Politehnica" University of Bucharest, Department of Automatic Control and Computer Science 313 Splaiul Independentei, Sector 6, 77206-Bucharest, ROMANIA
Gmail:
dan~karla.indinf.pub.ro
Abstract: In this paper, several methods to enlarge by parametrization the families of wavelets are proposed. The main goal is to prove that every remarkable classical wavelet can generate a class of wavelets, each one preserving the main characteristics of the original, but offering more mobility for time-frequency representations. Here, an important class is described: tight frames using wavelets. This class is used for apllications like: seismic signal processing, underwater acoustics, radar processing, image processing, automatic interpretation of images. Copyright © 19981FAC Keywords: parametric/orthogonal wavelets, continuous/numerable tight frames
1.
in an efficient way. It is not so easy to perform this requirement. One way is to use a class of wavelets mother rather than a single wavelet mother generating the affine family . The class can be generated by parametrization of a wavelet mother in such manner that its spectrum variation be controled.
INTRODUCTION
The wavelets history describes a sinuous evolu tion but very similar to of the other tools used in Signal Processing domain. After being introduced by Morlet (15 years ago, (1984-1985), (1992)), the wavelets were intensively studied and used in many applications of non stationary signal processing. The framework of these applications was constructed using a new approach, where the time-frequency/scale methods are used rather than classical Fourier methods (see Stefanoiu (1995)). A strong theory of wavelets was built in the first 10 years. Nowadays, this theory seems to be complete and the interest for wavelets begins to decrease. However, their properties show that they will continue long time to represen t a very useful tool in applications (see Meyer (1992)).
Notice that there is a difference between the terms (affine) family of wavelets and class of wavelets . A family is generated by the affine operator applied to the wavelet mother. Any element of the family can replace the wavelet mother and generate a new family, in order to focus on a special time-frequency zone. A class is generated by parametrization of the wavelet mother. In general, the parametrization is produced by a different technique as expressed when the affine operator is applied. Each wavelet of the class acts like a wavelet mother and a family ca be generated with . The spectrum splitting realized by the family is intimately dependent on the spectrum of the corresponding wavelet mother. Comparing to a family, a cla% offers another kind of time-frequency focus .
For example, in many applications (like underwater acoustics or radar monitoring), an important purpose is to perform automatic interpretations of timefrequency images. Such images can be obtained by means of the Wavelet Transform (WT) . Wavelets are used to focus on the irregularities of a non stationary signal, leading to large magnitudes of wavelet coefficients (the values of WT, in fact) . With these coefficents a surface can be represented over the timefrequency plane and, thus, the wavelets might produce an efficient detection of the relevant zones on this surface (i.e. of the small zones where an important quantity of signal energy is concentrated). The interpretation of the time-frequency image is strongly influenced by these zones. One important requirement to obtain this is adapting the spectrum of wavelet mother to the spectrum of the signal, otherwise the wavelets are unable to focus on the relevant time-frequency zones
This paper belongs to a series of articles devoted to parametric wavelets and describes the frames class, starting from several results of Daubechies (1988; 1990), and Mallat (1989).
2.
FRAMES BRIEF OVERVIEW
Let X be a vectorial space over the corps of scalars IK and call signals its vectors. We may assume, without loss of generality, that IK stands for IR or al. Let us consider that the structure of the space X is more
135
complex: a scalar product ( . , .) and a canonical norm 11 . 11 are defined so that ,1' becomes a Hilbert type space.
normalized, then A 2: 1, due to Bessel-Fourier inequality (1). Moreover, if A = 1, then the tight frame becomes an orthogonal basis.
Generally speaking, the main concept directly connected to a vectorial space is "the basis" and several general properties can be stated. If a family {eiLET ~ X has only normalized vectors (Ihll = 1, ViE I), then the following properties hold :
It can be proved that any frame of the space X is a system of generators, but its vectors are not linearly independent in general (see Daubechies (1990)) (the proof uses the concept of frame operator) . Thus, a frame is often called more than a basis and the bounds are considered to express its redundancy degree . This interpretation is more intuitive for tight frames .
• The inequality of Bessel-Fourier:
IIxll 2 ~
2:= I( x,
ei
)1 2
,
V x E .1' .
The frame bounds are very important in order to etablish the frame type, which affects tremendously how the space X is spanned by the frame .
(1 )
iET
In the case of B is available:
• The equivalence of Parseval-Plancherel : th e family is an orthonormal basis of X if and only if the identity below is verified :
II xl1 2 =
2:= I(x, e.i) 12 ,
Vx
E.1'.
the following spanning formula
A!B2:= (x, ei ) ei+ ·R.(x),
x=
(2)
> A,
VxEX , (5)
.ET
iET
where the remainder
By convention, the sum depicted in (1) or in (2) points to a classical sum, if I is at most a numerable set, but to an integral , if I is connex or if it has a num erabl e or finit e set of connex parts.
1I'R.( x )1I
It is often difficult to cons truct bases in .1' , so that only' particular systems of generators (not lin early independent) can be considered. Such systems are the fram es.
n is
bound ed :
~ 0 (~-1) Ilxll.
(6)
Here, "0(-)" points to Landau symbol: lim o(x) .%-0
= O.
X
This involves that an approximati ve reconstruction formula can be used for each signal of .1', as below:
Definition 1: (frame) x::::::
A family {ej} .ET ~ .l' f or which it exists a pair of numbers (A, B) such that 0 < A ~ B < 00 and:
All xll 2 ~
2:= I( x,
ei
)1 2 ~ Bllxl1 2 , Vx
A! B2:= (x ,
ei ) e i,
V x E ,1' .
(7)
JET
The accuracy of the approximation depends on the
EX, (3)
constant S
JET
~f
=
A
B _ A ' The constant S
.
IS
greater (the
bounds B and A are closer), the accuracy is better. is a frame in the space. .1'. The numbe.r·s A and B are called lover bOW1.d alld upper bOW1.d of the frame (respectively) .
Th e last remark is very useful. Let us take into consideration the limit case A = B and that lIedl = 1, ViE I . Then A 2: 1. We can even consider that the bound A points to an "orthogonality degree" of the tight frame. If A is close to 1, then the frame is near to orthogonality.
Definition 2: (tight frame) A frame having the same lower boulld as th e. upper bound (A = B) is a tight frame.
The reconstruction formul a using a tight frame (not necessarily normalized) is then exact:
For each tight frame { ej L ET ~ X the following identity holds:
x
= ~ 2:=
(x,
ei) e"
V x EX .
(8)
JET
Allxl12 =
2:= I( x , ej )1 2 ,
Vx
E .1' .
(4)
In this formula, only the bound A might cause serious problems in evaluation, but if A = 1, then the signal can be recovered in a very efficient way.
iET
This is very similar to Parse val- Plancherel identity (2) . Obviously, if the vectors of the tight frame are
Notice that the reconstruction formulae above are not
136
For theoretical purposes, some conventional formulae (Poisson type) are proposed, coming from (12):
unique. It is possible to find another scalar coefficients, {Ci};er, different from the former ones, so that:
x
=L
(9)
Ciei.
ier o
This is an important difference between frames and orthogonal bases. But, in practice, the uniqueness of the reconstruction formula is irrelevant. On the contrary, the accuracy of the recovered signal is very concerned and it is suitable to produce perfect reconstruction formulae like (8), even though they are not unique.
f(t)
=
J
fer) lio(t - r) dr
-00
f(t)
= 2171"
L
"It Em. .(13)
+00
J
fer)
e)n(t-r)
dr
ne~_oo
The frame can replace the basis in order to represent elements of the Hilbert space in a relaxed manner, as will be emphasized in the next sections . It is suitable to work at least with tight frames if orthogonal ba~es are difficult to obtain .
3.
+00
By convention, lio (.) is the Dirac distribution and lio [.) is the (discrete-time) unit impulse (the Kronecker symbol, in fact). One of the most important property with respect to the spaces L 1 (U.;") and L 2 (IK) is the separability , that is the existence of a numerable dense set. This involves the possibility to construct numerable bases or tight frames . One main problem of Signal Processing domain is to construct families of known continuoustime signals so that they be (unconditional) bases or/and tight frames (eventually, numerable ones) for the space L2(U{). The purpose is to produce efficient representations for at least each stable signal. For example, such families are constructed using wavelets.
SIGNALS FRAMEWORK
In Signal Processing domain, the continuous-tim e signals are usually vectors of the following Hilbert space: L 2 (IK) . This is the space of signals with finite energy (the square of the norm , in fact). It is well known that there are unstable signals with finite energy. Recall that the continuous-time stable signals are absolutely integrable and generate the space Ll(U().
Let IV be a stable, finite energy and continuous-time signal, called wavelet mother. Two types of wavelets families can be constructed using the wavelet mother :
The stable signals with finit e energy are very important in Signal Processing because frequency representations can be constructed with respect to them . The classical tool used to provide frequency representations is the Fourier Transform , defined as follows :
1. The continuous affine family: ,T,
'I' a ,b
t d!J - _1 r.;
()
va
,T, 'I'
(t - b) a
,
"ItEm., (14)
where a E m.,+ is the scale parameter and b E m. is the time translation parameter . 2. The discretized affine family:
It is possible to recover the original signal using its Inverse Fourier Transform:
\IJm ,n(t)d;Ja;;~IV(ao"'t-nbo), "ItEm. ,(15) where ao > 1 is the scale step, bo > 0 is the time tmllsiatioll step and n, m E :Z . This family was obtained from a continuous one, by "sampling" both the scale and the time axes, so that it is more natural to call it "discretized" instead of "discrete". In general, the word "discrete" is used in the framework of [2 (IK) space (discrete-time signals with finite energy) .
+00
f(t)
=
vk J
!(n)e+iOtdn,
VtEIR . (11)
-00 The equations (10) and (11) were extended for the space of distributions, Schwarz type. In theoretical approaches, one remarkable distribution is often used: Dirac distribution li o. Its Fourier Transforms (Direct and Inverse) are defined a<; follows : +00
1
..j2;
J
lio(t) e-)nt dt
= ~, v21r
- 00 +00
lio (t)
2.. 271"
J
e)nt
In the next sections, we show that if several properties are verified, tight frames of wavelets can be generated using these families.
4.
( 12)
dn,
Vt, n E IR .
CONTINUOUS TIGHT FRAMES
A continuous tight frame can be constructed if the wavelet mother verifies the following conditions:
-00
137
Notice that the continuous affine family (14) is not a frame . Every wavelet of the family must be multiplied with l/a in order to obtain a tight frame, according to the theorem above. In this case, any (stable) finite energy signal I can be represented using the continuous tight frame of wavelets. An analysis formula is obtained by projecting the signal on every wavelet IJi a,b of the continuous wavelets family (14) :
1. Admissibility:
0<
et d;j
( 16)
(et "+"
is an admissibility constant and the mark specifies that only positive pulsations are considered). 2. Spectrum parity :
W",.!(a, b)
1 = Fa
(17)
V'n E IR
The signal I can be recovered from its wavelet coeflicients, using an inversion formula (non unique , in general); thus, for all t E IR :
Thus, the admissibility condition (16) holds if and only if the wavelet mother has a hyperbolic decay:
I(t)
=
27r~'t
+ 00
(V't E IR, V' k E IN) :nd its spectrum is concentrated to high frequencies ~ 1Ji(0) = O. Usually, in the wavelets theory, not only 1Ji(0) is supposed to be zero , but also some derivatives are null in origin (;j;(1')(0) = 0, for p = 1, 2, 3, .. . ). This involves that the wavelet mother is a fast oscilating (high frequency) signal.
db W",.!(a, b) lJia,v(t) (22)
-00
This perfect reconstruction formula can be verified in a same way a.~ in the proof of Theorem 1 and the uniform convergence is due to (18). Thus, a continuous tight frame of L2(O{) wa.~ constructed using stable wavelets. Every signal with finite energy can be represented in this frame (according to analysis formula (21)) or it can be exactly recovered from its representation (according to synthesis formula (22)) .
Obviously, the parity of the spectrum implies that the values of IJi are all real or all imaginary (the real and the imaginary parts cannot be simul taneously non zero ).
This wa.~ a theoretical approach rather than practical, because, in practice, it is very difficult to compute the wavelets coefficients and to design an exact synthesis procedure . Therefore, in practice , it seems to be more interesting to construct numerable frames of (stable) wavelets.
Theorem 1: (Continuous tight frames) If the wavelet mother' '11 ve7'ifies the cOllditions (16) (j
+ 00
J~~ J o
is
(21 )
The function W"'.! is a Continuous Wavelet TransfO,.,/I and IW",.! I shows the continuous distribution of the signal's energy over the time-scale plane. The numbers W"•.!(a, b) are wavelet coefficients .
The restrictions above were defined in frequency domain, but it is also possible to give some interpretations in time domain.
{.!.a IJi ",b}
(t-b)
I(t) IJi -a- dt.
-00
(usually, all the values of wavelet mother are real, so that this condition is naturally satisfied) .
and (17), then the family:
J+OO
COll-
,,>O , bE/R
tinuous tight frame of L2(O() with the boulld:
5.
(19)
NUMERABLE TIGHT FRAMES
For the discretized affine family of wavelets (15), there are some general properties with respect to the frames, as proved Daubechies (1990).
For proving this result, it is sufficient to verify Definition 2. This can be straightforwardly done by computing the frame bound Thus, for all lEe (oq, the identity below holds:
A:
Here, a new approach is proposed . In order to obtain a particular numerable tight frame, the wavelet mother IJi is supposed to perform the conditions below: • The decay of IJi is fa.st enough to ensure that for all I E L 2 (D<) the series below is uniformly convergent:
In this evaluation, the uniform con vergence is ensured by the admissibility condi tion (16) and Dirac and Poisson formulae (12) and (13) are also involved.
I: I: (I, IJi m,n) IJi mE~ nE~
138
m ,n
ue.
(23)
This is an admissibility restriction and it implies that the string {( f , Wm," )} m,"E~ has finite energy:
L: L:
IU,
Wm,n
2
}1 < 00
mother is thus the impulse response of a pass band filter, easy to design. The second property is quite natural in practice; the left bound of the support must be strictly positive because the wavelet mother is a high frequency signal . Notice that the spectrum dimension and the logarithmic constant were choosen according to the Information Theory (see Daubechies (1990) or Stefanoiu (1995)).
(24)
.
mE~ nE~
Notice that every signal of L2(D\) is bounded and, moreover, it has a sufficient decay (for example, hyperbolic type) to ensure that its energy is finite . • The following identity holds:
L:
nE m:
I~ (a~O) 12 = { ~(ao)
12
kE~
=0
Theorem 2: (Numerable tight frames) If the wavelet mother W verifies the conditions (23) and (25), then the discretized affine family of wavelets is a numerable tight frame in L2(JK) with the bound:
.(25)
Here, '1'( ao) '" 0, 'Vao > 1. This restriction is Battle-Lemarie type (1987 ; 1988) . The sum above is practically constant with respect to n, but it depends on ao, following the mae.Y', When o = 0, this restriction implies that \j1(0) = 0 , which means that the wavelet mother is a high frequency signal, like ill the ca.<;e of continuous tight frames.
A
21r
= b;; '1'( ao) .
(29)
This statement can be proved by verifying Definition 2. Thus, the frame bound is straightforwardly evalu ated for all f E L 2 (D{) (taking into consideration that the sums below are convergent, due to the admissibility condition (23»:
In practice, W is constructed in such manner that tile conditions below are verified.
"" L...J "" L...J IU,
Wm,n}1 2 = b;;'P(ao)lIf h 112 .
(30)
1. The restriction of Battle-Lemarie type holds:
n E IR , (26)
In this evaluation, Dirac and Poisson formulae (12) and (13) are also involved.
where XA is the characteristic function of the set
If the map 'I' has the particular form expressed by (26), then the bound of the tight frame is:
'V
d;U A .. XA () a -
{I0
E
(L A (L!/. A .
A ___ 1r_ - bo In ao '
2. The spectrum of '11 is compactly supported; the length of the support equals 21rb o :
Supp (Here, 00
I~I = [no,
00
+ hbo 1
(31)
according to first Poisson formula (13) .
(27)
Every signal f E L2(lI{) can be represented in the tight frame, by projections on the wavelets Wm,n of the family (15):
> 0.)
First property is an aliasing formula , in fact. It is obvious that the support of the current term in the sum (26) can be expressed as follows:
W~(:j'°[m,nld~a~T
+00
J
f(t)W (a
o t-71bo) dt.(32) m
-00
The fUllction W~~/o is the Discretized Wavelet Transfonll and its values are the wavelet coefficients . This time, the time-scale plane is uncontinuously scanned: some kind of "hyperbolic" network covers it, that is every wavelet coefficient is localized in a zone bounded by 2 hyperbolic traces (left-right) and 2 horizontallinear traces (up-down) (see Stefanoiu (1995» . This is due to the fact that, for each m, a horizontal line is generated by varying 11 and, for each n, a hyperbolic curve is constructed by varying m . The energy of the signal is thus uncontinuously spreaded over this plane. Moreover, the product ba In aa must not exceed the constant 27r, according to Information Theory.
(Recall that ao > 1 and bo > 0.) This means that the interval [0, 00) is covered by the parti tion of all the supports. The partition might be disjoint or not, depending on what parameters no, ao and bo are choosen . The aliasing phenomenon is produced if at least 2 adjacent supports are not disjoint. If o 0 0 > 21rb ,then no aliasing is performed in (26). In ao - 1 . this case, the spectrum of the wavelet mother is com1
pactly supported and constant ( - - ) . The wavelet In ao
139
The identity (30) has an intuitive interpretation; it expresses a relationship between the (global) energy of the Discretized Wavelet Transform and the (global) energy of the signal f:
Proposition 1: Let W E L2(JK) be a stable wavelet verifying the properties (23) and (25). Let A be the bound of the tight frame it generates and fix 0' > O. If w'" is used instead of W as wavelet mother, then the corresponding discretized affine family of wavelets is also a tight frame with the bound:
1(f,wm,n)1 2=
L L m€2Z n€2Z
a b "[1n, = ""' ~ ""' ~ 1W'l'~f
nJ 12
271" ",(
A"
Ilfll 2 . (33)
= -A0' .
(36)
m€2Z n€2Z
In order to prove that the family of wavelets generated by wa is a tight frame, it is sufficient to compute the energy of the new Discretized Wavelet Transform, according to (33). This leads to the new bound (36) .
The reconstruction formula using the wavelet coefficients is very easy to express, if the admissibility condition (23) is also verified :
It is interesting to work with normalized wavelets. If the wavelet mother is normalized, then the entirely discretized affine family will be normalized, because the energy is not affected by the affine operation:
(34)
(To prove this identity, the same argument as in th e proof of Theorem 2 can b e used .) In this formula, only the bound A might cause serious problellls in evaluation, but if A = 1, then the signal can b e recovered in a very efficient way, a.~ the frame would be an orthogonal basis.
6.1
(37) This remark allows one to normalize the family of wavelets in a simple way, dividing every wavelet by the energy of wavelet mother . Thus, if the family is a tight frame with the bound A, then its normalized version is also a tight frame, as proves the equation below (for all f E L2(U{)) :
6. THE FRAMES CLASS The general case
In practice, it is not so difficult to construct the wavelet mother W in such a way that the discretized affine family it generates be a tight frame for L2(U() . It is often useful to control the redundancy degree of a tigh t frame not only by means of (LO and lio , bu t also by using an independent parameter. This is possible if a scaling operation is applied to the wavelet mother:
If Proposition 1 and equation (38) are combined, then a tight frame with normalized wavelets is available:
L L
'ne~
wa(t)
d;J foW(O't) ,
'It Elf{,
nEZl
I(
f,
~;'Ii' ) r= 0' 1I~1I2 IIfll2 . (39)
( 35)
A natural inequality holds in this where er > 0 is the scaling factor. N aturaliy, the energy of the wavelet mother is not affected by scaling operation (no matter what factor 0' > 0 is use{l) :
IIwll 2=
+00
Fourier property (1): 0'
dt
Iw"(t)l2 Ilt
= Ilw"112 ,
of the scaling factor is: er
~(n) =
_1_
fo
a.~
if a scaling opwa.~ applied on it:
~ (~) = ~l/"(n), 0'
vn
> 1,
1I~12 '
= 1I~12'
If Theorem 2
holds, then an orthonormal ba.~is of wavelets can be obtained by applying two operations on the wavelets of the frame: normalization with IIwll and scaling with
E 1R .
the factor 0'
=
211" 'P(ao)
b IIwll2 o
.
If the wavelet mother W generates a (numerable) tight frame with the bound A (depending on the discretization steps), then several parameters can control the
If er E (0,1), the scaling operation is a dilation; if er
< 0' :::;
Now, let us recall Parseval-Plancherel property (2). If the frame of wavelets is normalized, then it exists a scaling factor er so that the scaled frame becomes an orthonormal basis of wavelets. Obviously, the value
-00 but its Fourier Transform changes, eration with the inverse factor 1/0'
due to Bessel-
?:: 1. Thus, normally,
the scaling factor is bounded: 0
+00
JIw(t)12 = J
11~1I2
ca.~e,
the scaling operation is a contmction .
140
Thus, in this context, there are thonormal bases of wavelets with operation: it can always obtain an from a tight frame or a tight frame mal basis, by varying continuously
redundancy degree of the corresponding normalized frame: ao > 1, bo >
0, (\' E (0, 11:11
2 ].
Usually,
ao = 2 and bo is constant and proportional with the sampling rate of signals, so that only (\' can actually control the redundancy. Thus: • if Cl' '\. 0, then the frame redundancy increa.<;es; in particular, the great dilations applied to the wavelet mother increase the redundancy degree; • if
Cl'
/
11: 2 ' 11
7.
Cl'
= 11 ~12'
then the frame is an orthonormal
basis (completely unredulldant) .
6.2
CONCLUSION
In this paper, several techniques to construct classes of tight frames using wavelets were proposed. These techniques are founded on a set of preliminary natural requirements that the wavelet mother is supposed to perform, like admissibility and Battle-Lemarie type conditions. The parametrisation of wavelet mother (produced by scaling) can preserve its capacity to generate tight frames, if the scaling factor is correctly selected . Thus, an entire class of tight frames can be constructed and even orthonormal bases can be obtained .
then the frame redundancy de-
creases; in particular, if the wavelet mother is more and more contracted, then the frame becomes more and more unredundant and near to orthogonali ty; • if
no "isolated" orrespect to scaling orthonormal basis from an orthonorthe factor (\' .
The orthogonal case
The orthogonal wavelets have a spe c i~d behaviour if the scaling operator is applied in order La generate a parametric cla.5s.
REFERENCES
Battle G . (1987) . A Block Spin Construction of Ondelettes: 1. Lemarie Functions . Communications in Mathematical Physics 110, 601-615. Daubechies 1. (1988). OrthonorInal Ba.5es of Compactly Supported Wavelets - 1. First Approach. Communications on Pw·e and Applied Mathemutics XLI, 909-996. Dallbechies I. (1990). The Wavelet Transform, Time-Frequency Localisation and Signal Analysis. I.E.E.E. Transactions on Information Theory 36(9),961-1005. Groupillaud P., Grossmann A., Morlet J. (1984-1985) . Cycle-Octave and Related Transforms in Seismic Signal Analysis. Geoexploration 23, 85-102 . Lemarie P.G . (1988). Ondelettes it localisation exponentielle. Journal des Mathematiques Pures et Appliques 67, 227-236 . Mallat S. (1989) . A Theory for Multiresolution Signal Decomposition: the Wavelet Representation. I.E.E.E. Transactions on Pattern Analysis and Machine Intelligence 11(7), 674- 693 . Meyer Y. (1992). Wavelets - Their Pa.5t and Their Future. Proceedings of the 3Td International
The wavelet mother I1t is oT·thoyo7Zal if the discretized affine family (15) is orthonormal. This means that the orthogonality relation below holds : (40)
(It rn, n, p, q E ~). Such wavelets (compactly supported) were constructed by Daubechies (1988). Proposition 2: Let I1t E L2(JK) be a stable oT·thogoTwl w(welet mother and consider that the corresponding discndized affine family (15) verifies the admissibility condition (23) . Then this family is an orthonormal basis if and only if the following identity holds:
n
E fR.'
n=0
(41)
The proof of this resul t is not so ditficnl t to perform (the admissibility condition (23) is crucial). This proposition shows that in the orthogonal case, the map 'P used to express the condition (25) is constant with respect to ao, but it depends on bo .
Conference on Wavelets and Applications. 9-18 Toulouse, FRANCE . Morlet J., Grossmann A. (1992) . Wavelets - Ten Years Ago. Proceedings of the 3Td International Conference on Wave/ets and Applications. 3-7 Toulouse, FRANCE . ~tenl.noiu D. (1995). Signal Analysis by Ti1lleFrequency Methods . PhD thesis" Politehnica" University of Bucharest, Department of Automatic Control and Computer Science, ROMANIA.
If I1t is an orthogonal wavelet mother that generates a basis in L2(JK), then both admissibility conditions (24) and (23) are verified. Obviouosly, the basis is a tight frame with unitary bound and Proposition 2 shows that, in this case, the condition (25) (BattleLemarie type) is also verified. It follows that the wavelet mother 11t'" generates a tight frame having the bound IfCl' for all Cl' E (0,1]' according to Proposition 1.
141
PARAMETRIC WAVELETS: TIGHT FRAMES CLASS
DaIl $teIaIloiu "Politehnica" University of Bucharest, Department of Automatic Control and Computer Science 313 Splaiul Independentei, Sector 6, 77206-Bucharest, ROMANIA
Gmail:
dan~karla.indinf.pub.ro
Abstract: In this paper, several methods to enlarge by parametrization the families of wavelets are proposed. The main goal is to prove that every remarkable classical wavelet can generate a class of wavelets, each one preserving the main characteristics of the original, but offering more mobility for time-frequency representations. Here, an important class is described: tight frames using wavelets. This class is used for apllications like: seismic signal processing, underwater acoustics, radar processing, image processing, automatic interpretation of images. Copyright © 19981FAC Keywords: parametric/orthogonal wavelets, continuous/numerable tight frames
1.
in an efficient way. It is not so easy to perform this requirement. One way is to use a class of wavelets mother rather than a single wavelet mother generating the affine family . The class can be generated by parametrization of a wavelet mother in such manner that its spectrum variation be controled.
INTRODUCTION
The wavelets history describes a sinuous evolu tion but very similar to of the other tools used in Signal Processing domain. After being introduced by Morlet (15 years ago, (1984-1985), (1992)), the wavelets were intensively studied and used in many applications of non stationary signal processing. The framework of these applications was constructed using a new approach, where the time-frequency/scale methods are used rather than classical Fourier methods (see Stefanoiu (1995)). A strong theory of wavelets was built in the first 10 years. Nowadays, this theory seems to be complete and the interest for wavelets begins to decrease. However, their properties show that they will continue long time to represen t a very useful tool in applications (see Meyer (1992)).
Notice that there is a difference between the terms (affine) family of wavelets and class of wavelets . A family is generated by the affine operator applied to the wavelet mother. Any element of the family can replace the wavelet mother and generate a new family, in order to focus on a special time-frequency zone. A class is generated by parametrization of the wavelet mother. In general, the parametrization is produced by a different technique as expressed when the affine operator is applied. Each wavelet of the class acts like a wavelet mother and a family ca be generated with . The spectrum splitting realized by the family is intimately dependent on the spectrum of the corresponding wavelet mother. Comparing to a family, a cla% offers another kind of time-frequency focus .
For example, in many applications (like underwater acoustics or radar monitoring), an important purpose is to perform automatic interpretations of timefrequency images. Such images can be obtained by means of the Wavelet Transform (WT) . Wavelets are used to focus on the irregularities of a non stationary signal, leading to large magnitudes of wavelet coefficients (the values of WT, in fact) . With these coefficents a surface can be represented over the timefrequency plane and, thus, the wavelets might produce an efficient detection of the relevant zones on this surface (i.e. of the small zones where an important quantity of signal energy is concentrated). The interpretation of the time-frequency image is strongly influenced by these zones. One important requirement to obtain this is adapting the spectrum of wavelet mother to the spectrum of the signal, otherwise the wavelets are unable to focus on the relevant time-frequency zones
This paper belongs to a series of articles devoted to parametric wavelets and describes the frames class, starting from several results of Daubechies (1988; 1990), and Mallat (1989).
2.
FRAMES BRIEF OVERVIEW
Let X be a vectorial space over the corps of scalars IK and call signals its vectors. We may assume, without loss of generality, that IK stands for IR or al. Let us consider that the structure of the space X is more
135
complex: a scalar product ( . , .) and a canonical norm 11 . 11 are defined so that ,1' becomes a Hilbert type space.
normalized, then A 2: 1, due to Bessel-Fourier inequality (1). Moreover, if A = 1, then the tight frame becomes an orthogonal basis.
Generally speaking, the main concept directly connected to a vectorial space is "the basis" and several general properties can be stated. If a family {eiLET ~ X has only normalized vectors (Ihll = 1, ViE I), then the following properties hold :
It can be proved that any frame of the space X is a system of generators, but its vectors are not linearly independent in general (see Daubechies (1990)) (the proof uses the concept of frame operator) . Thus, a frame is often called more than a basis and the bounds are considered to express its redundancy degree . This interpretation is more intuitive for tight frames .
• The inequality of Bessel-Fourier:
IIxll 2 ~
2:= I( x,
ei
)1 2
,
V x E .1' .
The frame bounds are very important in order to etablish the frame type, which affects tremendously how the space X is spanned by the frame .
(1 )
iET
In the case of B is available:
• The equivalence of Parseval-Plancherel : th e family is an orthonormal basis of X if and only if the identity below is verified :
II xl1 2 =
2:= I(x, e.i) 12 ,
Vx
E.1'.
the following spanning formula
A!B2:= (x, ei ) ei+ ·R.(x),
x=
(2)
> A,
VxEX , (5)
.ET
iET
where the remainder
By convention, the sum depicted in (1) or in (2) points to a classical sum, if I is at most a numerable set, but to an integral , if I is connex or if it has a num erabl e or finit e set of connex parts.
1I'R.( x )1I
It is often difficult to cons truct bases in .1' , so that only' particular systems of generators (not lin early independent) can be considered. Such systems are the fram es.
n is
bound ed :
~ 0 (~-1) Ilxll.
(6)
Here, "0(-)" points to Landau symbol: lim o(x) .%-0
= O.
X
This involves that an approximati ve reconstruction formula can be used for each signal of .1', as below:
Definition 1: (frame) x::::::
A family {ej} .ET ~ .l' f or which it exists a pair of numbers (A, B) such that 0 < A ~ B < 00 and:
All xll 2 ~
2:= I( x,
ei
)1 2 ~ Bllxl1 2 , Vx
A! B2:= (x ,
ei ) e i,
V x E ,1' .
(7)
JET
The accuracy of the approximation depends on the
EX, (3)
constant S
JET
~f
=
A
B _ A ' The constant S
.
IS
greater (the
bounds B and A are closer), the accuracy is better. is a frame in the space. .1'. The numbe.r·s A and B are called lover bOW1.d alld upper bOW1.d of the frame (respectively) .
Th e last remark is very useful. Let us take into consideration the limit case A = B and that lIedl = 1, ViE I . Then A 2: 1. We can even consider that the bound A points to an "orthogonality degree" of the tight frame. If A is close to 1, then the frame is near to orthogonality.
Definition 2: (tight frame) A frame having the same lower boulld as th e. upper bound (A = B) is a tight frame.
The reconstruction formul a using a tight frame (not necessarily normalized) is then exact:
For each tight frame { ej L ET ~ X the following identity holds:
x
= ~ 2:=
(x,
ei) e"
V x EX .
(8)
JET
Allxl12 =
2:= I( x , ej )1 2 ,
Vx
E .1' .
(4)
In this formula, only the bound A might cause serious problems in evaluation, but if A = 1, then the signal can be recovered in a very efficient way.
iET
This is very similar to Parse val- Plancherel identity (2) . Obviously, if the vectors of the tight frame are
Notice that the reconstruction formulae above are not
136
For theoretical purposes, some conventional formulae (Poisson type) are proposed, coming from (12):
unique. It is possible to find another scalar coefficients, {Ci};er, different from the former ones, so that:
x
=L
(9)
Ciei.
ier o
This is an important difference between frames and orthogonal bases. But, in practice, the uniqueness of the reconstruction formula is irrelevant. On the contrary, the accuracy of the recovered signal is very concerned and it is suitable to produce perfect reconstruction formulae like (8), even though they are not unique.
f(t)
=
J
fer) lio(t - r) dr
-00
f(t)
= 2171"
L
"It Em. .(13)
+00
J
fer)
e)n(t-r)
dr
ne~_oo
The frame can replace the basis in order to represent elements of the Hilbert space in a relaxed manner, as will be emphasized in the next sections . It is suitable to work at least with tight frames if orthogonal ba~es are difficult to obtain .
3.
+00
By convention, lio (.) is the Dirac distribution and lio [.) is the (discrete-time) unit impulse (the Kronecker symbol, in fact). One of the most important property with respect to the spaces L 1 (U.;") and L 2 (IK) is the separability , that is the existence of a numerable dense set. This involves the possibility to construct numerable bases or tight frames . One main problem of Signal Processing domain is to construct families of known continuoustime signals so that they be (unconditional) bases or/and tight frames (eventually, numerable ones) for the space L2(U{). The purpose is to produce efficient representations for at least each stable signal. For example, such families are constructed using wavelets.
SIGNALS FRAMEWORK
In Signal Processing domain, the continuous-tim e signals are usually vectors of the following Hilbert space: L 2 (IK) . This is the space of signals with finite energy (the square of the norm , in fact). It is well known that there are unstable signals with finite energy. Recall that the continuous-time stable signals are absolutely integrable and generate the space Ll(U().
Let IV be a stable, finite energy and continuous-time signal, called wavelet mother. Two types of wavelets families can be constructed using the wavelet mother :
The stable signals with finit e energy are very important in Signal Processing because frequency representations can be constructed with respect to them . The classical tool used to provide frequency representations is the Fourier Transform , defined as follows :
1. The continuous affine family: ,T,
'I' a ,b
t d!J - _1 r.;
()
va
,T, 'I'
(t - b) a
,
"ItEm., (14)
where a E m.,+ is the scale parameter and b E m. is the time translation parameter . 2. The discretized affine family:
It is possible to recover the original signal using its Inverse Fourier Transform:
\IJm ,n(t)d;Ja;;~IV(ao"'t-nbo), "ItEm. ,(15) where ao > 1 is the scale step, bo > 0 is the time tmllsiatioll step and n, m E :Z . This family was obtained from a continuous one, by "sampling" both the scale and the time axes, so that it is more natural to call it "discretized" instead of "discrete". In general, the word "discrete" is used in the framework of [2 (IK) space (discrete-time signals with finite energy) .
+00
f(t)
=
vk J
!(n)e+iOtdn,
VtEIR . (11)
-00 The equations (10) and (11) were extended for the space of distributions, Schwarz type. In theoretical approaches, one remarkable distribution is often used: Dirac distribution li o. Its Fourier Transforms (Direct and Inverse) are defined a<; follows : +00
1
..j2;
J
lio(t) e-)nt dt
= ~, v21r
- 00 +00
lio (t)
2.. 271"
J
e)nt
In the next sections, we show that if several properties are verified, tight frames of wavelets can be generated using these families.
4.
( 12)
dn,
Vt, n E IR .
CONTINUOUS TIGHT FRAMES
A continuous tight frame can be constructed if the wavelet mother verifies the following conditions:
-00
137
Notice that the continuous affine family (14) is not a frame . Every wavelet of the family must be multiplied with l/a in order to obtain a tight frame, according to the theorem above. In this case, any (stable) finite energy signal I can be represented using the continuous tight frame of wavelets. An analysis formula is obtained by projecting the signal on every wavelet IJi a,b of the continuous wavelets family (14) :
1. Admissibility:
0<
et d;j
( 16)
(et "+"
is an admissibility constant and the mark specifies that only positive pulsations are considered). 2. Spectrum parity :
W",.!(a, b)
1 = Fa
(17)
V'n E IR
The signal I can be recovered from its wavelet coeflicients, using an inversion formula (non unique , in general); thus, for all t E IR :
Thus, the admissibility condition (16) holds if and only if the wavelet mother has a hyperbolic decay:
I(t)
=
27r~'t
+ 00
(V't E IR, V' k E IN) :nd its spectrum is concentrated to high frequencies ~ 1Ji(0) = O. Usually, in the wavelets theory, not only 1Ji(0) is supposed to be zero , but also some derivatives are null in origin (;j;(1')(0) = 0, for p = 1, 2, 3, .. . ). This involves that the wavelet mother is a fast oscilating (high frequency) signal.
db W",.!(a, b) lJia,v(t) (22)
-00
This perfect reconstruction formula can be verified in a same way a.~ in the proof of Theorem 1 and the uniform convergence is due to (18). Thus, a continuous tight frame of L2(O{) wa.~ constructed using stable wavelets. Every signal with finite energy can be represented in this frame (according to analysis formula (21)) or it can be exactly recovered from its representation (according to synthesis formula (22)) .
Obviously, the parity of the spectrum implies that the values of IJi are all real or all imaginary (the real and the imaginary parts cannot be simul taneously non zero ).
This wa.~ a theoretical approach rather than practical, because, in practice, it is very difficult to compute the wavelets coefficients and to design an exact synthesis procedure . Therefore, in practice , it seems to be more interesting to construct numerable frames of (stable) wavelets.
Theorem 1: (Continuous tight frames) If the wavelet mother' '11 ve7'ifies the cOllditions (16) (j
+ 00
J~~ J o
is
(21 )
The function W"'.! is a Continuous Wavelet TransfO,.,/I and IW",.! I shows the continuous distribution of the signal's energy over the time-scale plane. The numbers W"•.!(a, b) are wavelet coefficients .
The restrictions above were defined in frequency domain, but it is also possible to give some interpretations in time domain.
{.!.a IJi ",b}
(t-b)
I(t) IJi -a- dt.
-00
(usually, all the values of wavelet mother are real, so that this condition is naturally satisfied) .
and (17), then the family:
J+OO
COll-
,,>O , bE/R
tinuous tight frame of L2(O() with the boulld:
5.
(19)
NUMERABLE TIGHT FRAMES
For the discretized affine family of wavelets (15), there are some general properties with respect to the frames, as proved Daubechies (1990).
For proving this result, it is sufficient to verify Definition 2. This can be straightforwardly done by computing the frame bound Thus, for all lEe (oq, the identity below holds:
A:
Here, a new approach is proposed . In order to obtain a particular numerable tight frame, the wavelet mother IJi is supposed to perform the conditions below: • The decay of IJi is fa.st enough to ensure that for all I E L 2 (D<) the series below is uniformly convergent:
In this evaluation, the uniform con vergence is ensured by the admissibility condi tion (16) and Dirac and Poisson formulae (12) and (13) are also involved.
I: I: (I, IJi m,n) IJi mE~ nE~
138
m ,n
ue.
(23)
This is an admissibility restriction and it implies that the string {( f , Wm," )} m,"E~ has finite energy:
L: L:
IU,
Wm,n
2
}1 < 00
mother is thus the impulse response of a pass band filter, easy to design. The second property is quite natural in practice; the left bound of the support must be strictly positive because the wavelet mother is a high frequency signal . Notice that the spectrum dimension and the logarithmic constant were choosen according to the Information Theory (see Daubechies (1990) or Stefanoiu (1995)).
(24)
.
mE~ nE~
Notice that every signal of L2(D\) is bounded and, moreover, it has a sufficient decay (for example, hyperbolic type) to ensure that its energy is finite . • The following identity holds:
L:
nE m:
I~ (a~O) 12 = { ~(ao)
12
kE~
=0
Theorem 2: (Numerable tight frames) If the wavelet mother W verifies the conditions (23) and (25), then the discretized affine family of wavelets is a numerable tight frame in L2(JK) with the bound:
.(25)
Here, '1'( ao) '" 0, 'Vao > 1. This restriction is Battle-Lemarie type (1987 ; 1988) . The sum above is practically constant with respect to n, but it depends on ao, following the mae.Y', When o = 0, this restriction implies that \j1(0) = 0 , which means that the wavelet mother is a high frequency signal, like ill the ca.<;e of continuous tight frames.
A
21r
= b;; '1'( ao) .
(29)
This statement can be proved by verifying Definition 2. Thus, the frame bound is straightforwardly evalu ated for all f E L 2 (D{) (taking into consideration that the sums below are convergent, due to the admissibility condition (23»:
In practice, W is constructed in such manner that tile conditions below are verified.
"" L...J "" L...J IU,
Wm,n}1 2 = b;;'P(ao)lIf h 112 .
(30)
1. The restriction of Battle-Lemarie type holds:
n E IR , (26)
In this evaluation, Dirac and Poisson formulae (12) and (13) are also involved.
where XA is the characteristic function of the set
If the map 'I' has the particular form expressed by (26), then the bound of the tight frame is:
'V
d;U A .. XA () a -
{I0
E
(L A (L!/. A .
A ___ 1r_ - bo In ao '
2. The spectrum of '11 is compactly supported; the length of the support equals 21rb o :
Supp (Here, 00
I~I = [no,
00
+ hbo 1
(31)
according to first Poisson formula (13) .
(27)
Every signal f E L2(lI{) can be represented in the tight frame, by projections on the wavelets Wm,n of the family (15):
> 0.)
First property is an aliasing formula , in fact. It is obvious that the support of the current term in the sum (26) can be expressed as follows:
W~(:j'°[m,nld~a~T
+00
J
f(t)W (a
o t-71bo) dt.(32) m
-00
The fUllction W~~/o is the Discretized Wavelet Transfonll and its values are the wavelet coefficients . This time, the time-scale plane is uncontinuously scanned: some kind of "hyperbolic" network covers it, that is every wavelet coefficient is localized in a zone bounded by 2 hyperbolic traces (left-right) and 2 horizontallinear traces (up-down) (see Stefanoiu (1995» . This is due to the fact that, for each m, a horizontal line is generated by varying 11 and, for each n, a hyperbolic curve is constructed by varying m . The energy of the signal is thus uncontinuously spreaded over this plane. Moreover, the product ba In aa must not exceed the constant 27r, according to Information Theory.
(Recall that ao > 1 and bo > 0.) This means that the interval [0, 00) is covered by the parti tion of all the supports. The partition might be disjoint or not, depending on what parameters no, ao and bo are choosen . The aliasing phenomenon is produced if at least 2 adjacent supports are not disjoint. If o 0 0 > 21rb ,then no aliasing is performed in (26). In ao - 1 . this case, the spectrum of the wavelet mother is com1
pactly supported and constant ( - - ) . The wavelet In ao
139
The identity (30) has an intuitive interpretation; it expresses a relationship between the (global) energy of the Discretized Wavelet Transform and the (global) energy of the signal f:
Proposition 1: Let W E L2(JK) be a stable wavelet verifying the properties (23) and (25). Let A be the bound of the tight frame it generates and fix 0' > O. If w'" is used instead of W as wavelet mother, then the corresponding discretized affine family of wavelets is also a tight frame with the bound:
1(f,wm,n)1 2=
L L m€2Z n€2Z
a b "[1n, = ""' ~ ""' ~ 1W'l'~f
nJ 12
271" ",(
A"
Ilfll 2 . (33)
= -A0' .
(36)
m€2Z n€2Z
In order to prove that the family of wavelets generated by wa is a tight frame, it is sufficient to compute the energy of the new Discretized Wavelet Transform, according to (33). This leads to the new bound (36) .
The reconstruction formula using the wavelet coefficients is very easy to express, if the admissibility condition (23) is also verified :
It is interesting to work with normalized wavelets. If the wavelet mother is normalized, then the entirely discretized affine family will be normalized, because the energy is not affected by the affine operation:
(34)
(To prove this identity, the same argument as in th e proof of Theorem 2 can b e used .) In this formula, only the bound A might cause serious problellls in evaluation, but if A = 1, then the signal can b e recovered in a very efficient way, a.~ the frame would be an orthogonal basis.
6.1
(37) This remark allows one to normalize the family of wavelets in a simple way, dividing every wavelet by the energy of wavelet mother . Thus, if the family is a tight frame with the bound A, then its normalized version is also a tight frame, as proves the equation below (for all f E L2(U{)) :
6. THE FRAMES CLASS The general case
In practice, it is not so difficult to construct the wavelet mother W in such a way that the discretized affine family it generates be a tight frame for L2(U() . It is often useful to control the redundancy degree of a tigh t frame not only by means of (LO and lio , bu t also by using an independent parameter. This is possible if a scaling operation is applied to the wavelet mother:
If Proposition 1 and equation (38) are combined, then a tight frame with normalized wavelets is available:
L L
'ne~
wa(t)
d;J foW(O't) ,
'It Elf{,
nEZl
I(
f,
~;'Ii' ) r= 0' 1I~1I2 IIfll2 . (39)
( 35)
A natural inequality holds in this where er > 0 is the scaling factor. N aturaliy, the energy of the wavelet mother is not affected by scaling operation (no matter what factor 0' > 0 is use{l) :
IIwll 2=
+00
Fourier property (1): 0'
dt
Iw"(t)l2 Ilt
= Ilw"112 ,
of the scaling factor is: er
~(n) =
_1_
fo
a.~
if a scaling opwa.~ applied on it:
~ (~) = ~l/"(n), 0'
vn
> 1,
1I~12 '
= 1I~12'
If Theorem 2
holds, then an orthonormal ba.~is of wavelets can be obtained by applying two operations on the wavelets of the frame: normalization with IIwll and scaling with
E 1R .
the factor 0'
=
211" 'P(ao)
b IIwll2 o
.
If the wavelet mother W generates a (numerable) tight frame with the bound A (depending on the discretization steps), then several parameters can control the
If er E (0,1), the scaling operation is a dilation; if er
< 0' :::;
Now, let us recall Parseval-Plancherel property (2). If the frame of wavelets is normalized, then it exists a scaling factor er so that the scaled frame becomes an orthonormal basis of wavelets. Obviously, the value
-00 but its Fourier Transform changes, eration with the inverse factor 1/0'
due to Bessel-
?:: 1. Thus, normally,
the scaling factor is bounded: 0
+00
JIw(t)12 = J
11~1I2
ca.~e,
the scaling operation is a contmction .
140
Thus, in this context, there are thonormal bases of wavelets with operation: it can always obtain an from a tight frame or a tight frame mal basis, by varying continuously
redundancy degree of the corresponding normalized frame: ao > 1, bo >
0, (\' E (0, 11:11
2 ].
Usually,
ao = 2 and bo is constant and proportional with the sampling rate of signals, so that only (\' can actually control the redundancy. Thus: • if Cl' '\. 0, then the frame redundancy increa.<;es; in particular, the great dilations applied to the wavelet mother increase the redundancy degree; • if
Cl'
/
11: 2 ' 11
7.
Cl'
= 11 ~12'
then the frame is an orthonormal
basis (completely unredulldant) .
6.2
CONCLUSION
In this paper, several techniques to construct classes of tight frames using wavelets were proposed. These techniques are founded on a set of preliminary natural requirements that the wavelet mother is supposed to perform, like admissibility and Battle-Lemarie type conditions. The parametrisation of wavelet mother (produced by scaling) can preserve its capacity to generate tight frames, if the scaling factor is correctly selected . Thus, an entire class of tight frames can be constructed and even orthonormal bases can be obtained .
then the frame redundancy de-
creases; in particular, if the wavelet mother is more and more contracted, then the frame becomes more and more unredundant and near to orthogonali ty; • if
no "isolated" orrespect to scaling orthonormal basis from an orthonorthe factor (\' .
The orthogonal case
The orthogonal wavelets have a spe c i~d behaviour if the scaling operator is applied in order La generate a parametric cla.5s.
REFERENCES
Battle G . (1987) . A Block Spin Construction of Ondelettes: 1. Lemarie Functions . Communications in Mathematical Physics 110, 601-615. Daubechies 1. (1988). OrthonorInal Ba.5es of Compactly Supported Wavelets - 1. First Approach. Communications on Pw·e and Applied Mathemutics XLI, 909-996. Dallbechies I. (1990). The Wavelet Transform, Time-Frequency Localisation and Signal Analysis. I.E.E.E. Transactions on Information Theory 36(9),961-1005. Groupillaud P., Grossmann A., Morlet J. (1984-1985) . Cycle-Octave and Related Transforms in Seismic Signal Analysis. Geoexploration 23, 85-102 . Lemarie P.G . (1988). Ondelettes it localisation exponentielle. Journal des Mathematiques Pures et Appliques 67, 227-236 . Mallat S. (1989) . A Theory for Multiresolution Signal Decomposition: the Wavelet Representation. I.E.E.E. Transactions on Pattern Analysis and Machine Intelligence 11(7), 674- 693 . Meyer Y. (1992). Wavelets - Their Pa.5t and Their Future. Proceedings of the 3Td International
The wavelet mother I1t is oT·thoyo7Zal if the discretized affine family (15) is orthonormal. This means that the orthogonality relation below holds : (40)
(It rn, n, p, q E ~). Such wavelets (compactly supported) were constructed by Daubechies (1988). Proposition 2: Let I1t E L2(JK) be a stable oT·thogoTwl w(welet mother and consider that the corresponding discndized affine family (15) verifies the admissibility condition (23) . Then this family is an orthonormal basis if and only if the following identity holds:
n
E fR.'
n=0
(41)
The proof of this resul t is not so ditficnl t to perform (the admissibility condition (23) is crucial). This proposition shows that in the orthogonal case, the map 'P used to express the condition (25) is constant with respect to ao, but it depends on bo .
Conference on Wavelets and Applications. 9-18 Toulouse, FRANCE . Morlet J., Grossmann A. (1992) . Wavelets - Ten Years Ago. Proceedings of the 3Td International Conference on Wave/ets and Applications. 3-7 Toulouse, FRANCE . ~tenl.noiu D. (1995). Signal Analysis by Ti1lleFrequency Methods . PhD thesis" Politehnica" University of Bucharest, Department of Automatic Control and Computer Science, ROMANIA.
If I1t is an orthogonal wavelet mother that generates a basis in L2(JK), then both admissibility conditions (24) and (23) are verified. Obviouosly, the basis is a tight frame with unitary bound and Proposition 2 shows that, in this case, the condition (25) (BattleLemarie type) is also verified. It follows that the wavelet mother 11t'" generates a tight frame having the bound IfCl' for all Cl' E (0,1]' according to Proposition 1.
141