inverse problems
and theoretical imaging
I.M. Combes A. Grossmann
Ph. Tchamitchian (Eds.)
Wavelets
Time-Frequency Methods and Phase Space
Proceedings of the International Conference,
Marseille, France, December 14-18, 1987
With 88 Figures
Springer-Verlag Berlin Heidelberg New York
London Paris Tokyo Hong Kong
Professor Jean-Michel Combes
Professor Alexander Grossmann
Professor Philippe Tchamitchian
Centre National de la Recherche Scientifique
Luminy - Case 907, F-13288 Marseille Cedex 9, France
ISBN-13: 978-3-642-97179-2
e-ISBN-13: 978-3-642-97177-8
DOl: 10,1007/978-3-642-97177-8
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© Springer-Verlag Berlin Heidelberg 1989
Softcover reprint of the hardcover 1st edition 1989
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Preface
The last two subjects mentioned in the title "Wavelets" are so well established
that they do not need any explanations. The first is related to them, but a short
introduction is appropriate since the concept of wavelets emerged fairly recently.
Roughly speaking, a wavelet decomposition is an expansion of an arbitrary
function into smooth localized contributions labeled by a scale and a position parameter. Many of the ideas and techniques related to such expansions have existed
for a long time and are widely used in mathematical analysis, theoretical physics
and engineering. However, the rate of progress increased significantly when it was
realized that these ideas could give rise to straightforward calculational methods
applicable to different fields. The interdisciplinary structure (R.c.P. "Ondelettes")
of the C.N .R.S. and help from the Societe Nationale Elf-Aquitaine greatly fostered
these developments.
This conference was held at the Centre National de Rencontres Mathematiques
(C.I.R.M) in Marseille from December 14 to 18, 1987 and brought together an
interdisciplinary mix of participants. We hope that these proceedings will convey
to the reader some of the excitement and flavor of the meeting.
In the preparation of the conference we have benefited from the help and support of the following organisations: the Societe Mathematique de France and the
C.I.R.M.; the Universite Aix-Marseille IT, Faculte de Luminy; the Universite de
Toulon et du Var; the Conseil Regional Provence-Alpes-Cote d' Azur; the Laboratoire de Mecanique et Acoustique and Centre de Physique Theorique, both at the
C.N.R.S., Marseille. The company DIGILOG kindly provided the signal processor
SYTER for demonstration purposes.
The editors are extremely grateful to all of them, to the participants and to all
other people who helped in various ways to make this meeting a real success.
Marseille, December 1988
l.-M. Combes
A. Grossmann
Ph. Tchamitchian
(received: March 16, 1989)
v
In Memoriam
We have learned with shock the news of the sudden death of
Professor Franz B. Tuteur
His absence is keenly felt by those of us who had the privilege of knowing
him and working with him.
VI
Contents
Part I
Introduction to Wavelet Transforms
Reading and Understanding Continuous Wavelet Transforms
By A. Grossmann, R. Kronland-Martinet, and I. Morlet (With 23 Figures)
2
Orthonormal Wavelets
By Y. Meyer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
Orthonormal Bases of Wavelets with Finite Support - Connection with
Discrete Filters
By I. Daubechies (With 9 Figures) . . . . . . . . . . . . . . . . . . . . . . . . . .
38
Part II
Some Topics in Signal Analysis
Some Aspects of Non-Stationary Signal Processing with Emphasis on
Time-Frequency and Time-Scale Methods
By P. Flandrin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
Detection of Abrupt Changes in Signal Processing
By M. Basseville (With 1 Figure) . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
The Computer, Music, and Sound Models
By I.-C. Risset (With 2 Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
102
Part III
Wavelets and Signal Processing
Wavelets and Seismic Interpretation
By I.L. Larsonneur and I. Morlet (With 3 Figures)
126
Wavelet Transformations in Signal Detection
By F.B. Tuteur (With 4 Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
132
Use of Wavelet Transforms in the Study of Propagation of Transient
Acoustic Signals Across a Plane Interface Between Two Homogeneous
Media
By S. Ginette, A. Grossmann, and Ph. Tchamitchian (With 7 Figures) ..
139
VII
Time-Frequency Analysis of Signals Related to Scattering Problems in
Acoustics Part I: Wigner-Ville Analysis of Echoes Scattered by a Spherical
Shell
By J.P. Sessarego, J. Sageloli, P. Flandrin, and M. Zakharia
(With 4 Figures) ........... . . . . . . . . . . . . . . . . . . . . . . . . . . .. 147
Coherence and Projectors in Acoustics
By J.G. Slama ........................................
154
Wavelets and Granular Analysis of Speech
By J.S. Lienard and C. d' Alessandro (With 4 Figures) .............
158
Time-Frequency Representations of Broad-Band Signals
By J. Bertrand and P. Bertrand (With 2 Figures) .................
164
Operator Groups and Ambiguity Functions in Signal Processing
By A. Berthon ........................................
172
Part N
Mathematics and Mathematical Physics
Wavelet Transform Analysis of Invariant Measures of Some Dynamical
Systems
By A. Arneodo, G. Grasseau, and M. Holschneider (With 15 Figures) ..
182
Holomorphic Integral Representations for the Solutions of the Helmholtz
Equation
By J. Bros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
197
Wavelets and Path Integral
By T. Paul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
204
Mean Value Theorems and Concentration Operators in Bargmann and
Bergman Space
By K. Seip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
209
Besov Sobolev Algebras of Symbols
By G. Bohnke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
216
Poincare Coherent States and Relativistic Phase Space Analysis
By J.-P. Antoine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
221
A Relativistic Wigner Function Affiliated with the Weyl-Poincare Group
By J. Bertrand and P. Bertrand .............................
232
Wavelet Transforms Associated to the n-Dimensional Euclidean Group
with Dilations: Signal in More Than One Dimension
By R. Murenzi ........................................
239
Construction of Wavelets on Open Sets
By S. Jaffard (With 8 Figures) .............................
247
Wavelets on Chord-Arc Curves
By P. Auscher ........................................
253
VIII
Multiresolution Analysis in Non-Homogeneous Media
By R.R. Coifrnan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
259
About Wavelets and Elliptic Operators
By Ph. Tchamitchian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
263
Towards a Method for Solving Partial Differential Equations Using
Wavelet Bases
By V. Perrier (With 7 Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
269
Part V
Implementations
A Real-Time Algorithm for Signal Analysis with the Help of the Wavelet
Transform
By M. Holschneider, R. Kronland-Martinet, J. Morlet,
and Ph. Tchamitchian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
286
An Implementation of the "algorithme a trous" to Compute the Wavelet
Transform
By P. Dutilleux (With 7 Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . .
298
An Algorithm for Fast Imaging of Wavelet Transforms
By P. Hanusse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
305
SUbject Index
313
Index of Contributors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
315
IX
Introduction to Wavelet Transforms
Reading and Understanding Continuous Wavelet Transforms
A. Grossmann 1, R. Kronland-Martinet 2 , andJ. Morlet 3
1Centre de Physique Theorique, Section II, C.N.R.S.,
Luminy Case 907, F-13288 Marseille Cedex 09, France
2Faculre des Sciences de Luminy and Laboratoire de Mecanique
et d'Acoustique, C.N.R.S., 31, Chemin J. Aiguier,
F-13402 Marseille Cedex 09, France
3TRAVIS, c/o O.R.I.C. 371 bis, Rue Napoleon Bonaparte,
F-92500 Rueil-Malmaison, France
1. Introduction
One of the aims of wavelet transforms is to provide an easily interpretable visual
representation of signals. This is a prerequisite for applications such as selective
modifications of signals or pattern recognition.
This paper contains some background material on continuous wavelet transforms and a
description of the representation methods that have gradually evolved in our work. A related
topic, also discussed here, is the influence of the choice of the wavelet in the interpretation
of wavelets transforms. Roughly speaking, there are many qualitative features (in particularly
concerning the phase) which are independent of the choice of analyzing wavelet; however, in
some situations (such as detection of "musical chords") an appropriate choice of wavelet is
essential.
We also briefly discuss the finite interpolation problem for wavelet transforms with
respect to a given analyzing wavelet, and give some details about analyzing wavelets of
gaussian type.
2. Definitions
The continuous wavelet transform of a real signal s(t) with respect to the analyzing
wavelet g(t) (in general, g(t) is complex) may be defined as a function:
(2.1)
S(b,a)=
fa-fg ((t~b))S(t)
dt
(gdenotes the complex conjugate of g)
defined on the open "time and scale" half-plane H (b E R, a>O). We shall find it convenient to
use a somewhat unusual coordinate system on H, with the b-axis ("dimensionless time") facing
to the right and the a-axis ("scale") facing downward (Fig 2.1).
The a-axis faces downward since small scales correspond, roughly speaking, to high
frequencies, and we are used to seeing high frequencies above low frequencies.
2
The function (2.1) can also be written in terms of the Fourier transforms g(w), ~(w) of
sIt) and g(t). The expression is:
(2.1 ')
S(b,a)=
raf~ (aw) eibro g(w) dw
We impose on g the "admissibility condition"
cg=27tfl~(W)1 ~; <
00.
If
~(w)
is differentiable
(which we assume here), this implies:
~(O) = 0 i.e Jg(t)dt = 0
a- 1/2 gC~b) then
(2.1) can be written as a scalar
product: S(b,a) = <g(b,a)1 s>
The main motivation for the admissibility condition
convergence of:
is that it implies the (weak)
If we define g(b,a)(t) as g(b,a)(t) =
(2.2)
If
Ig(b.ab<g(b,a)1
d~~b
This operator (in the space L2(R ,dt) of signals of finite energy) is then easily shown to
be Cg 1, where 1 is the identity.
3. Graphical conventions
We want to display complex-valued functions such as (2.1) in a way which will allow us
to gather -visually- a certain amount of useful information about the signal sIt). Two
preliminary comments are in order here:
The qualitative (and visual) information gathered from our pictures is certainly not the
end of all desire of signal analysis. We believe however that it supplements in a non-trivial
way the information obtained by inspection of the signal itself, of its Fourier transform or of
one of its time-frequency representations such as Wigner-Ville. We shall not attempt here a
comparison of various methods, and refer e.g to [3].
The expression (2.1) depends manifestly on the choice of the analyzing wavelet g; as a
matter of fact, it is essentially symmetric in s and in g. In order to obtain full quantitative
information about s from its tranform S, we need to know the analyzing wavelet g. There are
however many features of the signal which can be seen on (2.1) and which are independent of
the choice of g. It will turn out that such features often involve the phase of the complexvalued function (2.1).
After these remarks, we get down to business:
The {b,a}-half-plane can be either displayed as in Fig. 2.1, or it can be mapped on the full
plane (b,-Iog(a)} (Fig. 3.1).
3.1
3
This second representation is indispensable if we want to display on a single picture
information in a wide range of scale parameters. Such is the case when one is concerned with
sound signals in the audible range, where a spread of 10 octaves is not excessive. A
disadvantage of these representation is that straight lines of the open {b,a} half-plane, if they
are not parallel or perpendicular to one of the axes, become exponential curves in the
logarithmic representation.
Voices:
We shall often consider restrictions of S(b,a) to fixed discrete values of the scale parameter.
Such a restriction S(b,aj) (aj fixed) is called a voice. In agreement with our preceding
discussion, two consecutive voices correspond to a fixed ratio
...!L. The most common
aj+1
situation is aj = aD 2jiv (j integer). where the integer v, the number of voices per octave,
defines a well-tempered scale in the sense of music. The value v=12 (well-tempered scale of
Western music) gives, in practice, a continuous picture.
How should the values of S(b,a) be represented:
Here we use two alternative representations. The first one, simpler to implement, consists in
plotting, say, the real part (or sometimes the modulus or the phase) of each voice, and place
such plots one above the other (see e.g Fig. 0). Such plots can carry quite detailed information,
but they do not give a truly two-dimensional picture of S(b,a).
A two-dimensional picture is provided by Figs. 1 to 16 (in color), and we shall now describe
the conventions used in these representations. On each one of the pictures 1 to 16, the {b,a}half-plane is represented in the logarithmic coordinates of Fig. 3.1. The quantities displayed
are the modulus and the phase of S(b,a); they are both shown on one and the same picture:
S(b,a) = I S(b,a) I ej<p(b,a)
(0
~
<p(b,a) < 2rr)
The modulus, I S(b,a) I , is color-coded in accordance with the palette visible on the pictures.
The actual coding is done as follows: Let x =
Isl!~xl ~
1 be the value of lSI, normalized to its
maximum within the picture. The "true colors" (as distinguished from black or white) are used
in an interval sat.min ~ x ~sat.max; the values of sat.min and sat.max are chosen for each
picture so as to emphasize the features of interest. They are given, together with other
relevant information, in Table 1. If x<sat.min, then the color is white (small modulus). If
x>sat.max, then the color is black (large modulus saturation). Notice that sat.max can be
greater than 1; this only means that black will never appear on the pictures. The progression of
"true colors" can be seen in the horizontal stripes of Fig. 1. In that picture, for any fixed b, the
function a -> IS(b,a)1 is a gaussian. The asymetry of the stripes is due to the use of log(a) as a
coordinate.
The local value of the phase <p(b,a) is given by the density of black dots on the picture. As
the simplest example, consider Fig.1. From left to right, in a period, one can follows an
increase in density, corresponding to a regular increase of the phase of S(b,a) from 0 to 2rr.
When the phases reaches 2rr, it is wrapped around to the value 0; these lines where the density
of dots drops abruptly to zero are clearly visible on the pictures and will play an important
role in the interpretation, as highly visible lines of constant phase.
We have adopted one further convention in order to increase the legibility of the
pictures. If at a point {b,a} the modulus I S(b,a) I is smaller than a cutoff cutoff.ph , we decree
that the phase shall not be represented (Le., equivalently from the point of view of the
graphical representation, that it shall be set equal to zero). The value of cutoff.ph may be
equal to sat.min as in Fig. 1, but this is not necessary. For instance, in Fig. 2, one has cutoff.ph
< sat.min, so that the lines of constant phase can be followed also for very small values of the
modulus.
The first five columns of Table 1 give, from left to right, the number of the picture, the
quantities displayed (e.g. in fig. 13 and 14 only the modulus is shown), and the values of the
parameters just discussed. The sixth column (duration) gives the time interval that would
correspond to the picture if the continuous signal were sampled 32000 times per second.
4
It is sometimes convenient to display a slightly modified form of (2.1) namely the
function: T(b,a) =
Ja
S(b,a) =
a- 1
fg
((t~b))
s(t) dt . This is the function shown if the column
"normal." contains 1/a.
Finally, the last column gives the number of voices per octave and the total number of
voices.
4. Localization
a) Locality in time
The correspondance (2.1) is, in general, non-local. The value of S(b,a) at a point
H depends on s(t) for all t.
Assume however that g(t) vanishes outside some interval [tmin ,tmaxl . We may ask two
questions:
1) Which domain of the {b,a} plane can be influenced by the value of s(t) at to (i.e in
an arbitrary small neighbourhood of a point to)? The answer is obvious from (2.1). The "domain
of influence" of the point to is the cone to - b E aA with vertex at the point b=to on the edge of
the {b,a} half-plane (Fig. 4.1).
{b,a}
E
In logaritmic representation, the b-axis is sent to infinity, and the cone of Fig (4.1) becomes
the domain shown in Fig. 4.2.
b
a
Figure 4.1
Log a
Log a = 0 --I-----+-~-------1~
Figure 4.2
The second question is: which values s(t) can influence the transform S(bo,ao) at a given
point of the {b,a}-plane?
The same equation as above, namely to E aoA + bo , now gives an interval determined by a
cone facing upward from {bo,ao} (Fig.4.3).
5
b
Figure 4.3
a
b) Locality in frequency
We now change the assumptions about the wavelet g, and assume that its Fourier
transform ~(Ol) vanishes outside an interval r=(Olmin(g),Olmax(g)). We ask now the same questions
as above:
1) Which domain of the {b,a} plane can be influenced by the value of a Fourier component
g(OlO) ? There is no loss in generality in supposing 000>0. The answer comes now from (2.1'); if
we restrict g(Ol) to a small neighbourhood of 000, then (2.1') vanishes if Oloa is not in a small
neighbourhood of (Olmin(g),Olmax(g)). So the domain of influence of a Fourier component ~(0l0) of
the signal is the horizontal strip:
Olmin(g)
Olmax(g)
- - - < a < - - - of the (b,a}-half-plane (Fig.4.4).
000
000
•
b
w
Figure 4.4
2) Which Fourier components of the signal are felt at the point {bo,ao} of the (b,a}-halfplane? The answer is : The components ~(Ol) such that:
Olmin(g)
Olmax(g)
---<00< - - -
a
a
5. Covariance, progressivity
A very simple but fundamental property of the continuous wavelet transform is its
covariance with respect to shifts and dilations of the signal.
Fix the analyzing wavelet g. If S(b,a) is the transform of s(t), then S(b-to ,a) is the
1
t
b a .
transform of s(t-to), and S( ~;):" ) IS the transform of
s(~) (1..>0).
."fi:
6
- Complex monochromatic signals:
The covariance under time shifts has an immediate consequence. Assume for a moment
that s(t) is an eigenvector of the shift operator :
s(t-to) = A.(to) s(t), which can be satisfied only by s(t) = exp(iwot). Then, S(b,a) is satisfies
S(b-to , a) = A.(to) S(b,a), which means that S(b,a) is of the form:
(5.1 )
S(b,a) = exp(iwob) f(a)
(f(a) may be complex-valued).
The frequency of a complex monochromatic signal can be read off from the phase of any
restriction of its wavelet transform to a horizontal line a=constanl. This fact is independent
of the choice of the wavelet and a consequence of nothing else than shift covariance.
The function f(a) in (5.1) can be calculated. If we put &(00) = 6(00 - roo) in (5.1), we obtain
from (2.1'):
(5.2)
S(b,a) =
-Va
exp(iwob)
9' (awo)
The modulus of (5.2) is constant along lines of constant scale, and varies as
-Va ~
(awo) along line of constant time.
A "spectral line" in the signal gets translated in a horizontal pattern in the transform,
with constant modulus and linear phase.
- Real monochromatic signal : Progressive wavelets
The above discussion does not apply if the complex monochromatic signal is replaced by
a real monochromatic signal such as cos(wot). The transform (2.1 ') of this function is:
(5.3)
1
'2 exp(iwob)
-
9' (awo) + '21 exp(-iwob) -9'
(-awo)
The modulus of this function is not constant on lines of constant scale because of
interference between the two summands. This is a serious disadvantage for graphical
interpretation.
It is, however, clear that this problem does not arise if the one the terms of (5.3)
vanishes, i.e if §(w) vanishes on a half-axis. If §(w) = 0 for 00 < 0, we shall say that g(t) is a
progressive wavelet. That is, a progressive wavelet is defined as a complex-valued function
that satisfies the admissibility condition of Sec.2 and does not have Fourier components on the
negative frequency axis.
All the examples shown in this paper were calculated with progressive wavelets. Figs 1
and 2 show the transform of a real monochromatic signal with respect to two different
progressive wavelets. The relevant feature here is that the colored domains, which represent
the modulus, are horizontal strips of constant width. On any horizontal line, the phase varies
linearly. Its rate of variation is the frequency of the signal, which can thus be accurately
measured from the phase picture. This fact is independent of the choice of analyzing wavelet.
The frequency of a monochromatic signal can also be read off from the modulus of its
wavelet transform. If g is progressive, this modulus is 1 g(awo) I. We see that the relationship
here is less intuitive and that it depends on the choice of g.
- Homogeneous signals
A function f(t) is said to be homogeneous of order ex at the point t=o (ex arbitrary, it may
even be complex) if:
f(A.t)= A.(1f(t)
7
In other words. f is an eigenfunction of the dilation operator. Since dilations do not
interchange the positive and negative axis. the natural example (analogous to complex
exponential for shift operators) lives on one side of 0:
fit) = {
~a
(t>O)
(t$O)
It is convenient to introduce a normalization factor. and define:
__
1_ta
(bO)
1)
u(i(t) = {
q:+
(t$O)
Considered as a distribution. ut(t) is entire analytic in its dependence on ex (see e.g [4]).
If ex is a negative integer. then ut(t)
is a derivative of the S-function:
(n=1.2 •..... ).
The wavelet transform of an homogeneous signal is fully determined by its restriction to
any line a=const.
6. Reproduci ng kernel
The transform (2.1) is a correspondance between the function sit) of one variable and the
function S = Lgs of two variables. It is reasonable to expect that S is not arbitrary.
One set of equations satisfied by S is deduced easily from the expression (2.2) for the
identity operator in L2.
Since
S(b.a) = < g(b,a) Is>. we have:
1
S(b.a) = Cg
i.e
S(b.a) =
f
If
< g(b,a) I g(b',a') > da'db'
~ < g(b'.a') Is>
pg(b.a;b·.a') S(b'.a·)
pg(b.a;b'.a·) =
(6.1 )
=
~
9
d~~2b'
where
< g(b,a) I g(b',a') >
~g ~fg (a't-~+b')
rK"J e '(bl - b')1 a'
"\Ja
1 _
= Cg
g(t) dt
A am .Co
g (~\:j(m)dm
Equation (6.1) says that pg is the reproducing kernel for the space of functions S(b,a) that
are wavelet transforms. with respect to g. of signals sit) of finite energy. We shall also say
that Pg is the reproducing kernel associated to g.
From expression (6.1) one sees that Ipg (b.a;b· .a·)1 attains its maximal value when
{b.a}={b·.a·}. With the wavelets that we use. Ipg(b.a;b·.a')1 decays very fast when. say. {b'.a'}
moves away from {b.a}. In other words. for fixed {bo.ao}, pg(bo.ao; .•. ) is a function on H that
is localized around {bo.ao}. In the following section. we shall use finite families of such
functions to obtain local approximation to S(b.a).
As an illustration. we give here the scalar product < g(b,a) I g(O. 1) > where
g(t)=eictexp(- ~ t
cp
2} The phase of this scalar product is:
bc(1+a)
= (1 +a2)
while its modulus is:
8
m=
_ {2rta
(
!.. b 2+c( 1-a)2)
-\J ~a2 exp - 2
(1+a2)
This function and a function of the type < g(b,a) I g(b O,1/2) > are displayed on Fig.9.
Another example is shown on Fig.10. This example will be discussed later.
If F(b,a) is an arbitrary function on the {b,a}-half-plane, such that:
If
ff
IF(b,a)i2
d~~b
<
(finite energy). then the function
00
pg(b,a;b',a') F(b',a')
da'db'
~
is the transform of some signal s(t) of finite energy.
7. Local approximations to S(b,a)
It is known that a wavelet transform S(b,a) is fully determined by its values on a
suitable grid of the {b,a}-half-plane; this grid depends on the choice of the analyzing wavelet
(see the article of I. Oaubechies in these proceedings). We are now caught in a dilemma: On the
one hand, the continuous function S(b,a) has many desirable properties (full covariance with
respect to shifts and dilation, simple interpretation, etc.), on the other hand, computing and
storing this function on very fine grids is clearly wasteful of computer time and memory.
We shall now derive a very simple "local interpolation" formula which does the
following:
We start with n arbitrary points Pl={bl,alL ..... ,Pn={bn,an} of the {b,a}-half-plane. We
assume that the points are distinct; Pj 7' Pj if i7'j. We assume that an analyzing wavelet g is
given, and that the wavelet transform S=Lgs of a signal s is known at the points Pj; the value
of S at Pj is the complex number Sj.
S(bj,aj) = Sj
(i=1 ... n)
We shall approximate S(b,a) (on an appropriate compact subset of arguments b,a) by a
linear combination of the functions ej(b,a) = pg(bj,aj;b,a) introduced in the preceding section:
n
(7.1) Sappr(b,a) = LYj ej(b,a)
j=1
We shall determine the coefficients Yj by the requirement that Sappr should take the
"correct" values Sj = S(bj,aj) at the points Pj (i=1 .. n).
It should be stressed that the basic "Ansatz" (7.1) can be wildly wrong as an
approximation of S, e.g if {b,a} is taken to be "far away" from all the points Pj. Notice however
that at such a point all the functions ej(b,a) are very small, by the basic concentration
properties of our wavelets (and consequently of reproducing kernels). If the points Pj are not
spaced too far from each other (e.g if they are adjacent elements of a grid giving rise to a good
frame) and if P={a,b} is chosen inside the convex hull of these points, the approximation (7.1)
can be excellent.
The determination of the coefficients Yj is easy. The interpolation conditions are:
n
Sj = Sappr(bj,aj) = LYj ej(bj,aj)
j=1
n
=
L, Yj
j=l
pg(bj,aj;bj,aj)
n
=L,AjjYj
j=1
Where A = (Ajj) is the n by n Gram matrix:
1
(bi,ai) (b',a')
Ajj = pg(bj,aj;bj,aj) = - < g i g J J >
Cg
(Notice the order of i and j) which is known to be hermitean and positive definite.
9
Introducing the inverse B
A-1, we find:
n
'Yj =
L
B jj S j
j=1
and the final local approximation formula:
(7.2)
Sappr(b,a) =
n
n
;=1
j=1
L L
with ej(b,a) = pg(bj,aj;b,a)
Sj = S(bj,aj)
ej(b,a) Bjj S j
(i=1 .. n)
0=1 .. n)
Covariance of the interpolation-approximation formu la.
The result (7.2) would be of little use if the matrices A and B had to be re-calculated
whenever the interpolation nodes P1 ... P n are changed. This is in fact not necessary. The formula
(7.2) is invariant with respect to the two basic families of transformations which define the
natural geometry of the {b,a}-half-plane H : the time shifts and the rescalings.
In order to visualize the content of these statements, it is useful to think of H in the
linear (rather than logarithmic) representation of sec. 3. A time shift (by to E R) of the points
{P1· ... Pn}={{b1,a1} .... {bn,an}} brings them into the "congruent" family of points
{{b1+tO,a1} ... {bn+to,an}}. Similarly, a re-scaling (by bO, and at the point b=O on the boundary
a=O of the half-plane) brings them into the "congruent" family of points {{Ab1,Aaj) .... {Abn,Aan}}.
The re-scaling at a different point b=bo of the boundary can be written in terms of the time
shifts and of the re-scalings at b=O; such general re-scalings together with time-shifts are
the most general transformations in the natural geometry of H.
The covariance statement is then:
If one transforms simultaneously
(i) the interpolation nodes P1 ... Pn
(ii) the points P={b,a}
by one of the geometrical transformations of H, then the only item to be changed in (7.2) are
the numbers Sj (which will of course correspond to different values of S(b,a) ).
This remark is useful in the practical implementation of the "fleshing out" of the
transform starting from its skeleton on a grid.
8. Admissible and almost progressive gaussians
Gaussians (shifted in time, in frequency and re-scaled) have many properties which
recommend them as analyzing wavelets. They have the best p<>ssible simultaneous
concentration in time and in frequency. The set of their finite linear combinations is closed
under Fourier transform, pointwise multiplication and convolution. The scalar product of any
two members of this set is given by an explicit formula. They are among the very few classes
of functions where the transition from one to more dimensions is immediate.
We have, however, to reconcile this praise of gaussians with our requirements that an
analyzing wavelet be admissible and progressive. While a finite linear combination of
gaussians may be admissible, no such combination can be progressive, because the tail of any
gaussian extends to infinity.
In the words of W.C. Fields, the time has come to take the gaussian bull by the tail and
face the situation.
Progressivity and admissibility may of course be enforced by the simple expedient of
"cutting the tail" of a gaussian in the frequency space. This is however best done on a linear
combination of gaussians, at a point where this linear combination has a zero of sufficently
high order. We now describe the construction of such linear combinations, which also keep
some of the good properties described above.
1) We shall start by introducing a linear combination of gaussians of different widths,
all centered at x=O, that vanishes at ~, where c is a preassigned positive number.
It is useful to require that our linear combination be invariant under Fourier transform
(like the basic gaussian). Define:
10
ho(x) = exp( -t x2 )
Choose a number A>O, and consider the dilated gaussian with the same L2-norm:
(i")
(DAho)(x) = A- 1/2 h o
Then the Fourier transform of DAh o is D11Ah o.
Consequently, for any real y, the function
ho(x)- y[(DAho)(x) + (D 11Aho)(x)] is invariant under Fourier transform, real and symmetric
under x -> -x. We can make it vanish at X=!C by choosing:
ho(c)
(8.1 )
(i}
y=-------~~-------
A- 1/2 h o
A1/2ho(AC)
We define consequently
h1(c;A;x) = ho(x) _y(A- 1I2 hO(i") +
(8.2)
A1/2
hO(AX))
where y is given by (8.1). Since h 1(c;A;x) = hdc;A· 1;x), there is no loss of generality in assuming
that A~1. We have h 1(tc;1 ;x) = 0.
The n-th derivative of h1 (C;A;X) with respect to x is:
h(~)(c;A;x)
= (-1)n[He n(X)h o(X) - y (A· n·tHe n(A· 1X)h o(i") +
An+~Hen(AX)ho(AX))]
Here the Hen(x) are modified Hermite polynomials:
Hen(x) = 2· n/2
H{~) , and
the Hn the usual Hermite polynomials.
It is now easy to construct functions hn = hn(C;A1 ,.... An ;x) that:
(i) are finite linear combinations of gaussians
(ii) are invariant under Fourier transform : An = hn
(iii) have a zero of order n at te.
Take n distinct numbers A1, ... ,An > 1, and define hn as the determinant
h1(C;A1 ;x) ........ h1(C;An;x)
h; (C;Ap) ........ h; (C;An;X)
2) Consider now the function
gn(C;A1 .... An;x) = eicx hn(C;A1···. An;x)
by the above, the Fourier transform of gn has a zero of n-th order at ~=O. With reasonable
values of C,A1 , .... ,An the function gn will be practically progressive, and suitable for numerical
work.
- Gaussian chirp
The wavelets that we just considered are (cosmetic) improvements on the basic wavelet
eiCX.exp(-t x2 ) introduced by one of us a while ago. If c~5 this wavelet is practically admissible
11
and progressive. Its "instantaneous frequency· (derivative of the phase with respect to time)
is independent on the time x.
We shall now contruct a related wavelet with instantaneous frequency that increase
linearly in time. Such "gaussian chirps" are known in various fields.
In order to save time, we shall not repeat here the discussion of the preceding section
concerning the enforcement of strict admissibility and progressivity.
We consider now the wavelet:
(8.3)
21
1
.
kx 2
elCXexp(i
exp (-2" x2)
where c is as above, and bO. This k is the rate of increase of instantaneous frequency c+kx.
Some of the examples described in the next section have been computed with the wavelet (8.3).
- The "two-humped" wavelet
The humps here are in frequency space, and the wavelet is of the form:
(8.4)
gh(X) = (exp(ic 1x) + exp(ic2X)) exp(-} x2)
with Fourier transform:
(8.4')
~h(~) = exp(-t (~-C1)2) + exp(-t (~-C2)2)
where both C1 and C2 are sufficiently large so that (8.4) is practically progressive. The
motivation for introducing this and similar wavelets is the detection, in the signal, of
contributions that correspond to a given "chord". This is a variation on the "matched filter"
theme. The transform of a monochromatic signal of frequency C3 with respect to the wavelet
(8.4) is:
(8.5)
S(b,a) =
~
exp(ibc3) (§h(a(c3-C1)) +
~h(a(C3-c2)))
The modulus of S is the same as the modulus of the transform associated to the sum of two
monochromatic components, taken with respect to the one-humped wavelet (9.1). However, in
contrast to that case, the rate of change of the phase is independent of the scale parameter a.
Examples of "octave detection" will be seen on Figs. 13 and 14.
9. Exampl es
Fig. 0 : The signal to be analyzed is shown at the bottom of the picture. It corresponds to
the sound "e" of the word "person". The total duration is about 23 ms. Just above the signal, one
can see its reconstruction from seven voices, with the help of the one-dimensional
reconstruction formula; (see e.g [9]). The real part of the voices are shown in the upper part of
the picture. There is one voice per octave. The highest voice is centered at 4000Hz, and the
lowest one at 62.5Hz.
Fig. 1 : This figure is a representation of the wavelet transform of the real
monochromatic signal discussed in Sec. 5. One can see on it the features described here:
horizontal strips of constant modulus, and phase in step with the phase of the signal. The
analyzing wavelet is the modulated gaussian :
e icx exp(-t x2) with c=5.0.
Fig. 2 : This figure shows the transform of a monochromatic wave with respect to the
wavelet (8.3). It should be compared with Fig. 1 where a monochromatic wave is analyzed with
the help of the wavelet (9.1). The important difference between the two pictures lies in the
behaviour of lines of constant phase. These lines are straight and vertical in Fig. 1 and
parabolas with horizontal axis in Fig. 2. A simple calculation shows that the maximum modulus
of the transform is obtained at points where these parabolas have vertical tangent. This can
also be seen clearly on the figure.
The phase pictures made with the help of wavelets (8.3) can thus be used for the
detection of spectral lines in a signal.
(9.1)
12