A Theory for Multiresolution Signal Decomposition: The Wavelet Representation

A Theory for Multiresolution Signal Decomposition: The Wavelet Representation
674
IEEE TRANSACTIONSON PATTERNANALYSISAND MACHINE INTELLIGENCE.VOL. II, NO. 7. JULY 1%')
A Theory for Multiresolution Signal Decomposition:
The Wavelet Representation
STEPHANE
Abstract-Multiresolution
representations are very effective for analyzing the information
content of images. We study the properties of
the operator which approximates a signal at a given resolution. We
show that the difference of information
between the approximation
of
a signal at the resolutions 2’ + ’ and 2jcan be extracted by decomposing
this signal on a wavelet orthonormal
basis of L*(R”).
In LL(R ), a
wavelet orthonormal
basis is a family of functions ( @ w (2’~ n)) ,,,“jEZt, which is built by dilating and translating a unique function
t+r(xl. This decomposition defines an orthogonal multiresolution
representation called a wavelet representation.
It is computed with a pyramidal algorithm based on convolutions with quadrature mirror lilters. For images, the wavelet representation
differentiates
several
spatial orientations. We study the application of this representation to
data compression in image coding, texture discrimination
and fractal
analysis.
Index Terms-Coding,
fractals, multiresolution
pyramids,
ture mirror filters, texture discrimination,
wavelet transform.
quadra-
I. INTRODUCTION
N computer vision, it is difficult to analyze the information content of an image directly from the gray-level
intensity of the image pixels. Indeed, this value depends
upon the lighting conditions. More important are the local
variations of the image intensity. The size of the neighborhood where the contrast is computed must be adapted
to the size of the objects that we want to analyze [41].
This size defines a resolution of reference for measuring
the local variations of the image. Generally, the structures
we want to recognize have very different sizes. Hence, it
is not possible to define a priori an optimal resolution for
analyzing images. Several researchers [ 181, [3 11, [42]
have developed pattern matching algorithms which process the image at different resolutions. For this purpose,
one can reorganize the image information into a set of
details appearing at different resolutions. Given a sequence of increasing resolutions (rj )jEz, the details of an
image at the resolution rj are defined as the difference of
information between its approximation at the resolution rj
and its approximation at the lower resolution rj _ ,
I
Manuscript received July 30. 1987: revised December 23. 1988. This
work was supported under the following Contracts and Grants: NSF grant
IODCR-84 1077 1. Air Force Grant AFOSR F49620-85-K-0018.
Army
DAAG-29-84-K-0061.
NSF-CERiDC82-19196
Ao2. and DARPAiONR
ARPA N0014-85-K-0807.
The author is with the Department of Computer Science Courant Institute of Mathematical Sciences. New York University. New York, NY
10012.
IEEE Log Number 8928052.
G. MALLAT
A multiresolution decomposition enables us to have a
scale-invariant interpretation of the image. The scale of
an image varies with the distance between the scene and
the optical center of the camera. When the image scale is
modified, our interpretation of the scene should not
change. A multiresolution representation can be partially
scale-invariant if the sequence of resolution parameters
( ‘; ),E~ varies exponentially. Let us suppose that there exists a resolution step cy E R such that for all integers j, rj
= oJ. If the camera gets 01times closer to the scene, each
object of the scene is projected on an area cy2times bigger
in the focal plane of the camera. That is, each object is
measured at a resolution cx times bigger. Hence, the details of this new image at the resolution olj correspond to
the details of the previous image at the resolution cy/ +I.
Resealing the image by cx translates the image details
along the resolution axis. If the image details are processed identically at all resolutions, our interpretation of
the image information is not modified.
A multiresolution representation provides a simple hierarchical framework for interpretating the image information [22]. At different resolutions, the details of an image generally characterize different physical structures of
the scene. At a coarse resolution, these details correspond
to the larger structures which provide the image “context”. It is therefore natural to analyze first the image details at a coarse resolution and then gradually increase the
resolution. Such a coarse-to-fine strategy is useful for pattern recognition algorithms. It has already been widely
studied for low-level image processing such as stereo
matching and template matching [ 161, [ 181.
Burt [5] and Crowley [8] have each introduced pyramidal implementation for computing the signal details at
different resolutions. In order to simplify the computations, Burt has chosen a resolution step Q! equal to 2. The
details at each resolution 2’ are calculated by filtering the
original image with the difference of two low-pass filters
and by subsampling the resulting image by a factor 2 ‘.
This operation is performed over a finite range of resolutions. In this implementation, the difference of low-pass
filters gives an approximation of the Laplacian of the
Gaussian. The details at different resolutions are regrouped into a pyramid structure called the Laplacian pyramid. The Laplacian pyramid data structures, as studied
by Burt and Crowley, suffer from the difficulty that data
at separate levels are correlated. There is no clear model
0162-8828/89/0700-0674$01
.OO 0 1989 IEEE
675
MALLAT. THEORY FOR MULTIRESOLUTION SIGNAL DECOMPOSlTlON
which handles this correlation. It is thus difficult to know
whether a similarity between the image details at different
resolutions is due to a property of the image itself or to
the intrinsic redundancy of the representation. Furthermore, the Laplacian multiresolution representation does
not introduce any spatial orientation selectivity into the
decomposition process. This spatial homogeneity can be
inconvenient for pattern recognition problems such as texture discrimination [2 11.
In this article, we first study the mathematical properties of the operator which transforms a function into an
approximation at a resolution 2 J. We then show that the
difference of information between two approximations at
the resolutions 2 J + ’ and 2 J is extracted by decomposing
the function in a wavelet orthonormal basis. This decomposition defines a complete and orthogonal multiresolution representation called the wavelet representation.
Wavelets have been introduced by Grossmann and Morlet
[ 171 as functions $(x) whose translations and dilations
(A rl/(.= - t)) ,u__,)ER+xR can be used for expansions of
L’(R)
functions.
Meyer
[35]
showed
that
there exists wavelets +(x) such that ( J2/ 1c/(2 ‘x k))i,,I;)Ez? is an orthonormal basis of L’( R ). These bases
generalize the Haar basis. The wavelet orthonormal bases
provide an important new tool in functional analysis. Indeed, before then, it had been believed that no construction could yield simple orthonormal bases of L’ (R ) whose
elements had good localization properties in both the spatial and Fourier domains.
The multiresolution approach to wavelets enables us to
characterize the class of functions II/(x) E L*(R) that
generate an orthonormal basis. The model is first described for one-dimensional signals and then extended to
two dimensions for image processing. The wavelet representation of images discriminates several spatial orientations. We show that the computation of the wavelet representation may be accomplished with a pyramidal
algorithm based on convolutions with quadrature mirror
filters. The signal can also be reconstructed from a wavelet representation with a similar pyramidal algorithm. We
discuss the application of this representation to compact
image coding, texture discrimination and fractal analysis.
In this article, we omit the proofs of the theorems and
avoid mathematical technical detals. Rather, we try to illustrate the practical implications of the model. The mathematical foundations are more thoroughly described in
w31.
A. Notation
Z and R denote the set of integers and real numbers
respectively. L’( R ) denotes the vector space of measurable, square-integrable one-dimensional functions f (x ).
Forf(x)
E L*(R) and g(x) E L’(R),
the inner product
off(x) with g(x) is written
The norm of f(x)
in L’ (R ) is given by
I/ flj*
= 11,
1.f’(41*
du.
We denote the convolution of two functionsf(x)
and g(x) E L*(R) by
f*
g(x)
= (f(u) * g(u)) (x>
+CC
J-W
g(x - u) du.
=I’
The Fourier transform off(x)
and is defined by
f(w)
E L* (R )
= II,f(x)
E L* (R ) is written p( w )
Cwr dx.
Z* (Z ) is the vector space of square-summable sequences
L* (R*) is the vector space of measurable, square-integrable two dimensional functionsf(x,
y). Forf(x,
y) E
L*(R*) and g(x, y) E L’(R’),
the inner product off(x,
y) with g(x, y) is written
( f(X, y), g(x, ?1,> = 11,” sI_ m.
The Fourier transform of f(x,
P(% w,.) and is defined by
&,,
w,.) = i+m s+mf(x,
-cc -m
II.
MULTIRESOLUTION
Y> g(x, Y) dx dJ.
y) E L*( R2) is written
y) ,-‘(wl’+uixl
dx dy.
TRANSFORM
In this section, we study the concept of multiresolution
decomposition for one-dimensional signals. The model is
extended to two dimensions in Section IV.
A. Multiresolution Approximation of L’(R )
Let A?, be the operator which approximates a signal at
a resolution 2 j. We suppose that our original signal f(x)
is measurable and has a finite energy: f(x) E L’(R).
Here, we characterize AZ, from the intuitive properties that
one would expect from such an approximation operator.
We state each property in words, and then give the equivalent mathematical formulation.
1) A*, is a linear operator. If A?,f(x) is the approximation of some function f(x) at the resolution 2 ‘, then
A2,f(x) is not modified if we approximate it again at the
resolution 2 ‘. This principle shows that A*,0 Azi = A2 ,.
The operator A,, is thus a projection operator on a particular vector space V,, C L’ (R ). The vector space V2, can
be interpreted as the set of all possible approximations at
the resolution 2 J of functions in L’ (R) .
2) Among all the approximated functions at the resolution 2”, A?,f(x) is the function which is the most similiar tof(x).
676
IkEE TRANSACTIONS ON PATTkRN ANALYSIS AND MACHINE INTELLIGENCE. VOL. Il. NO. 7. JULY I%39
vg(x>
E v21,
jjdx)
-f(x)/1
2
II&Sb)
-S(x)/I.
(1)
Hence, the operator A*, is an orthogonal projection on the
vector space VI,.
3) The approximation of a signal at a resolution 2”’
contains all the necessary information to compute the same
signal at a smaller resolution 2 J. This is a causality property. Since Al, is a projection operator on VI, this principle is equivalent to
VjEZ,
v*, c v*,i1.
(2)
4) An approximation operation is similar at all resolutions. The spaces of approximated functions should thus
be derived from one another by scaling each approximated function by the ratio of their resolution values
Vj E Z,
f(x)
E v,,
* f(2x)
E V2,’ 1.
(3)
5) The approximation AZ/f(x) of a signal f(x) can be
characterized by 2.’ samples per length unit. Whenf(x)
is translated by a length proportional to 2 I, A?, f (x ) is
translated by the same amount and is characterized by the
same samples which have been translated. As a consequence of (3), it is sufficient to express the principle 5)
for the resolution j = 0. The mathematical translations
consist of the following.
l
Discrete characterization:
There exists an isomorphism Z from V, onto Z* (Z ) .
(4)
l
Translation of the approximation:
vk E Z, A,fx(x)
= A,f(x
l
- k), whereh(x)
=f(x
- k).
(5)
Translation of the samples:
Z(A,f(-~))
= (my,)i~z * Z(Mi(x))
= (vn>,,,.
(6)
6) When computing an approximation of f(x) at resolution 2 ‘, some information aboutf (x) is lost. However,
as the resolution increases to + 00 the approximated signal
should converge to the original signal. Conversely as the
resolution decreases to zero, the approximated signal contains less and less information and converges to zero.
Since the approximated signal at a resolution 2 ’ is equal
to the orthogonal projection on a space V2,, this principle
can be written
lim V2, = ;” V?, is dense in L2 (R )
,++CC
,=-'72
(7)
lim
J--m
(8)
and
V,, =
77 V,, = (0).
J= -cc
We call any set of vector spaces ( V21),Ez which satisfies
the properties (2)-(8) a multiresolution approximation of
L’ (R ). The associated set of operators A*, satisfying l)6) give the approximation of any L’ ( R ) function at a res-
olution 2 I. We now give a simple example of a multiresolution approximation of L’ ( R ).
Example: Let V, be the vector space of all functions of
L’( R ) which are constant on each interval ] k, k + 1 [,
for any k E Z. Equation (3) implies that V2, is the vector
space of all the functions of L’ (R ) which are constant on
each interval ] k2 -j, (k + 1)2-’ [, for any k E Z. The
condition (2) is easily verified. We can define an isomorphism Z which satisfies properties (4), (5), and (6) by associating with any function f(x)
E V, the sequence
( CY~)~~~such that (Ye equals the value of f(x) on the interval ] k, k + 1 [. We know that the vector space of
piecewise constant functions is dense in L? (R ). Hence,
+Oa
we can derive that U I’,, is dense in L’ (R >. It is clear
+CO
J’-m
n
VI,=
(0) , so the sequence of vector spaces
,=-cc
( Vr,),EZ is a multiresolution
approximation of L’(R).
Unfortunately, the functions of these vector spaces are
neither smooth nor continuous, making this multiresolution approximation rather inconvenient. For many applications we want to compute a smooth approximation. In
Appendix A, we describe a class of multiresolution
approximations where the functions of each space Vz , are n
times continuously differentiable.
We saw that the approximation operator A?, is an orthogonal projection on the vector space Vz,. In order to
numerically characterize this operator, we must find an
orthonormal basis of Vz,. The following theorem shows
that such an orthonormal basis can be defined by dilating
and translating a unique function 4 (x).
Theorem 1: Let ( V2,),Ez be a multiresolution approximation of L’ (R ). There exists a unique function 4 (x) E
L’ (R ), called a scaling function, such that if we set
&!(x)
= 2’$(2’x)
forj E Z. (the dilation of 4(x) by
2’), then
that
( &?
&,(x
- 2m’n)),rGz is an orthonormal basis of I’2 1.
n
(9)
Indications for the proof of this theorem can be found
in Appendix B. The theorem shows that we can build an
orthonormal basis of any V,, by dilating a function $ (x)
with a coefficient 2 ’ and translating the resulting function
on a grid whose interval is proportional to 2-‘. The functions &,(x)
are normalized with respect to the L’ (R )
norm. The coefhcient J2-’ appears in the basis set in order to normalize the functions in the L’( R ) norm. For a
given multiresolution approximation ( V21)JEz, there exists
a unique scaling function 4(x) which satisfies (9). However, for different multiresolution
approximations,
the
scaling functions are different. One can easily show that
the scaling function corresponding to the multiresolution
described in the previous example is the indicator function of the interval [0, I]. In general, we want to have a
smoother scaling function. Fig. 1 shows an example of a
continuously differentiable and exponentially decreasing
scaling function. Its Fourier transform has the shape of a
677
MALLAT: THEORY FOR MULTIRESOLUTION SIGNAL DECOMPOSITION
ok) ;
Hence, we can rewrite Af,f:
A&f = ((f(u)
-5
(a)
ao,
i
1 t
i
B. Implementation
\
0 i __iJ
I
-70
-Jc
0
7T
w
10
(b)
Fig. 1. (a) Example of scaling function 4(x). This function ia computed
in Appendix A. (b) Fourier transform i(w). A scaling function ia a lowpass filter.
low-pass filter. The corresponding multiresolution
approximation is built from cubic splines. This scaling function is described further in Appendix A.
The orthogonal projection on V2, can now be computed
by decomposing the signal f(x) on the orthonormal basis
given by Theorem 1. Specifically,
vf(x)
E L’(R),
= 2-”
AZ/~(X)
!iY (f(u),
,1=-C=
- 2-‘n))
&,(x
- 2-‘n).
( 10)
The approximation of the signalf(x)
at the resolution 2 ‘,
A,,f(x),
is thus characterized by the set of inner products
which we denote by
A&f
= (( f(u),
hi(u
- 2y’n))),lFZ.
(11)
A!, f is called a discrete approximation off(x) at the resolution 2 .‘. Since computers can only process discrete signals, we must work with discrete approximations. Each
inner product can also be interpreted as a convolution
product evaluated at a point 2:‘n
(f(u),
&,(u
= (f(u)
* bi(
-u))
of a Multiresolution
Transform
qb2,(x - 2-‘n)
2-‘-l
,z
m
=
(&,(u
- 2:/n),
&,+I(u
- 2-‘-‘k))
- 2-‘+k).
(13)
By changing variables in the inner product integral, one
can show that
2-‘-I(
&,(u
- 2-‘n),
&,+~(u
- 2-‘-‘k))
$(u - (k - 2n))).
(14)
When computing the inner products off(x)
sides of (13) we obtain
with both
= (42-1(u),
( f (UL +2,(u
= ,=g,
-
2+n)
>
(5% I(U), +(u - (k - 2n)))
. (f(u),
42,+,(~
- 2-‘-‘k)).
Let H be the discrete filter whose impulse response is
given by
- 2-44)
*cc
em
f(u) bi(u
=s
(12)
(2-/n))ntz.
In practice, a physical measuring device can only measure a signal at a finite resolution. For normalization purposes, we suppose that this resolution is equal to 1. Let
Aff be the discrete approximation at the resolution 1 that
is measured. The causality principle says that from Af’f
we can compute all the discrete approximations A:, f for j
< 0. In this section. we describe a simple iterative algorithm for calculating these discrete approximations.
Let (V2,),d be a multiresolution approximation and
4 (x) be the corresponding scaling function. The family
is an orof functions (m
+*,+1(x - 2P”P’k)),,,
thonormal basis of V2,,+1. We know that for any n E Z, the
function $*,(x - 2-‘n) is a member of V?, which is included in V2,+ I. It can thus be expanded in this orthonormal basis of V,,, , :
* &t+,(x
&,(u
-u)>
Since 4(x) is a low-pass filter, this discrete signal can be
interpreted as a low-pass filtering off(x)
followed by a
uniform sampling at the rate 2 ‘. In an approximation operation, when removing the details off(x)
smaller than
2-‘, we suppress the highest frequencies of this function.
The scaling function 4 (x) forms a very particular lowpass filter since the family of functions (fl
&,
(x - 2Pin))REz is an orthonormal family.
In the next section we show that the discrete approximation off(x)
at the resolution 2 ’ can be computed with
a pyramidal algorithm.
X
5
0
* h(
- 2-/n) du
(2-/n).
Vn E Z, h(n)
= (4+(u),
6(u - n)).
(15)
Let fi be the mirror filter with impulse response i; (n ) =
h( -n). By inserting (15) in the previous equation, we
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE. VOL. II, NO. 7. JULY IYXY
678
obtain
(f(u),
Al-:fd
- 2-/n) >
h(u
= kjYm h(2n - k) (f(U),
(f12,Tl(u - 2-‘-h)).
(16)
,..‘.
A2mzfd ;""'~'~~~"~~,.....,,,~,
Equation (16) shows that Ai,f can be computed by convolving A:,+ of with fi and keeping every other sample of
the output. All the discrete approximations A,d,f, forj <
0, can thus be computed from Affby
repeating this process. This operation is called a pyramid transform. The
algorithm is illustrated by a block diagram in Fig. 5.
In practice, the measuring device gives only a finite
number of samples: A:f = (a,,), i ,, I N. Each discrete signal Af,,f(j
< 0) has 2 ‘N samples. In order to avoid border problems when computing the discrete approximations Af,f, we suppose that the original signal Af’f is
symmetric with respect to n = 0 and n = N
(a-,,
if-N<n<O
i %.Y-,I
if0
(4, =
and
14’(x)/
= 0(x-“).
=
.".."'
_,,_
___.
----. ,.
?-;f d _.’-...x..-_.___,___
;. ,____.. ‘.-.,....
.----.I,_...._
*,/*.
Y-----1...-
A Lfd
3-
5:
.__. 13:
.-.,:
152
'.,,---zoo
',
25,
(a)
E h(n)ep”‘“.
,1=--m
Fig. 2. (a) Discrete approximations A: ,fat the resolutions I. l/2. I /4.
I /8. I / 16. and l/32. Each dot gives the amplitude of the inner product
( ,f( u), & (u - 2 -‘?I ) ) depending upon 2-‘,1. (b) Continuous approximations A,,f(.r) at the resolutions 1. l/2. I /4. I /8. I /l6. and I /32.
These approximations are computed by interpolating the discrete approximations with (IO).
H(w)
satisfies the following
iH(
IH(,)/’
= 1 and
+ (H(w
h(n)
(17)
two properties:
= O(n-‘)
at infinity.
+ a)/’ = 1.
(17a)
(17b)
Conversely let H( w ) be a Fourier series satisfying (17a)
and (17b) and such that
IH(
# 0
for
w E [0, 7r/2].
(17c)
The function defined by
J(w)
The following theorem gives a practical characterization
of the Fourier transform of a scaling function.
Theorem 2: Let 4(x) be a scaling function, and let H
be a discrete filter with impulse response h(n)
=
($2 I(U), 4(u - n)>. Let H(w) be the Fourier series
defined by
H(o)
'...,,..
< II < 0.
If the impulse response of the filter fi is even ( fi = H ),
each discrete approximation At,fwill
also be symmetric
with respect to n = 0 and n = 2-/N. Fig. 2(a) shows the
discrete approximated signal Af,f of a continuous signal
f(x) at the resolutions 1, l/2,
l/4,
l/8,
l/16, and
l/32.
These discrete approximated signals have been
computed with the algorithm previously described. Appendix A gives the coefficients of the filter H that we used.
The continuous approximated signals AIlf(x)
shown in
Fig. 2(b) have been calculated by interpolating the discrete approximations with (10). As the resolution decreases, the smaller details off(x)
gradually disappear.
Theorem 1 shows that a multiresolution approximation
; b),ez is completely characterized by the scaling function 4 (x). A scaling function can be defined as a function
4(x) E L’(R)
such that, for all j E Z, (J2-’ &,
(x - 2-in)),,,, is an orthonormal family, and if VI, is the
vector space generated by this family of functions, then
is a multiresolution approximation of L’( R ). We
(V&z
also impose a regularity condition on scaling functions.
A scaling function 4 (x) must be continuously differentiable and the asymptotic decay of 4(x) and 4’ (x) at infinity must satisfy
(45(x)1 = O(F)
A
.'.
= ,g,
H(2-“w)
(18)
is the Fourier transform of a scaling function.
n
Indication for the proof of this theorem are given in
Appendix C. The filters that satisfy property (17b) are
called conjugare filters. One can find extensive descriptions of such filters and numerical methods to synthesize
them in the signal processing literature [lo], [36], [40].
Given a conjugate filter H which satisfies (17a)-( 17c), we
can then compute the Fourier transform of the corre-
679
MALLAT: THEORY FOR MULTIRESOLUTION SIGNAL DECOMPOSITION
h(n)
sponding scaling function with (16). It is possible to
choose H(o) in order to obtain a scaling function 4 (x)
which has good localization properties in both the frequency and spatial domains. The smoothness class of
4 (x) and its asymtotic decay at infinity can be estimated
from the properties of H( w ) [9]. In the multiresolution
approximation given in the example of Section II-A, we
saw that the scaling function is the indicator function of
the interval [0, 11. One can easily show that the corresponding function H ( w ) satisfies
H(w)
= e-‘” cos
0
0.6
t
4
III. THE WAVELET REPRESENTATION
As explained in the introduction, we wish to build a
multiresolution representation based on the differences of
information available at two successive resolutions 2 J and
2JC’. This section shows that such a representation can
be computed by decomposing the signal using a wavelet
orthonormal basis.
A. The Detail Signal
Here, we explain how to extract the difference of information between the approximation of a function f(x)
at the resolutions 2 J + ’ and 2 ‘. This difference of information is called the detail signal at the resolution 2,‘. The
approximation at the resolution 2j’ ’ and 2’ of a signal
are respectively equal to its orthogonal projection on
Vz,+ I and Vz,. By applying the projection theorem, we can
easily show that the detail signal at the resolution 2’ is
given by the orthogonal projection of the original signal
on the orthogonal complement of V,, in Vz,+ 1. Let 02, be
this orthogonal complement, i.e.,
V2,,
02, 0 v2, = V2,+,
To compute the orthogonal projection of a functionf(x)
we need to find an orthonormal basis of O2 ,. Much
like Theorem 1, Theorem 3 shows that such a basis can
be built by scaling and translating a function $ (x).
Theorem 3: Let (V?,)jEZ be a multiresolution vector
space sequence, 4 (x) the scaling function, and H the corresponding conjugate filter. Let +(x) be a function whose
Fourier transform is given by
on 02,,
$(w)=G;
4
i)
;
0
with
I
I
-20
I
-10
,.
0
I....
10
20
n
(a)
Appendix A describes a class of symmetric scaling
functions which decay exponentially and whose Fourier
transforms decrease as 1 /w”, for some n E N. Fig. 3
shows the filter H associated with the scaling function
given in Fig. 1. This filter is further described in Appendix A.
O?, is orthogonal to
r
G(w) = e-la H(w + T) .
(19)
:g
;'---I
0
-J
-n
LI
,
1
-2
0
2
I..
7Tm
(b)
Fig. 3. (a) Impulse response of the filter H associated to the scaling func;ion shown in Fig. I .-The coefficients of this filter are given in Appendix
A. (b) Transfer function H(w) of the filter H.
Let &,(x)
2,‘. Then
(@
= 2’$(2’x)
$2/(X
denote the dilation of +5(x) by
- 2-Jn)),*pzis an orthonormal basis of 02,
and
is an orthonormal basis of L* (R ).
H
$(x) is called an orthogonal wavelet.
Indications for the proof of this theorem can be found
in Appendix D. An orthonormal basis of 02, can thus be
computed by scaling the wavelet $(x) with a coefficient
2 J and translating it on a grid whose interval is proportional to 2-j. The wavelet function corresponding to the
example of multiresolution given in Section II-A is the
Haar wavelet
1
rc/(x) =
i
-1
L 0
ifOlx<i
if$lx<
1
.
otherwise
This wavelet is not even continuous. In many applications, we want to use a smooth wavelet. For computing a
wavelet, we can define a function H(w) which satisfies
the conditions (17a)-(17c) of Theorem 2, compute the
corresponding scaling function 4 (x) with equation (18)
680
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and the wavelet $(x) with (19). Depending upon choice
of H( w), the scaling function 4(x) and the wavelet $(x)
can have good localization both in the spatial and Fourier
domains. Daubechies [9] studied the properties of 6 (x)
and $(x) depending upon H( w ). The first wavelets found
by Meyer [35] are both C” and have an asymptotic decay
which falls faster than the multiplicative inverse of any
polynomial. Daubechies shows that for any II > 0, we
can find a function H(w) such that the corresponding
wavelet $(x) has a compact support and is n times continuously differentiable [9]. The wavelets described in
Appendix A are exponentially decreasing and are in C”
for different values of n. These particular wavelets have
been studied by Lemarie [24] and Battle 131.
The decomposition of a signal in an orthonormal wavelet basis gives an intermediate representation between
Fourier and spatial representations. The properties of the
wavelet orthonormal bases are discussed by Meyer in an
advanced functional analysis book [34]. Due to this double localization in the Fourier and the spatial domains, it
is possible to characterize the local regularity of a function f(-r) based on the coefficients in a wavelet orthonormal basis expansion [25]. For example, from the
asymptotic rate of decrease of the wavelet coefficients, we
can determine whether a function f(x) is n times differentiable at a point x0. Fig. 4 shows the wavelet associated
with the scaling function of Fig. 1. This wavelet is symmetric with respect to the point x = l/2. The energy of
a wavelet in the Fourier domain is essentially concentrated in the intervals [ -2~, -ir] U [T, 27r].
Let PO:, be the orthogonal projection on the vector
space O?,. As a consequence of Theorem 3, this operator
can now be written
cm
PO,f(x) = 2-’ ,,=F, (f(u), IcZl(U- 2-‘n) >
. l&,(.x - 2-/n).
(20)
PO2f(.~) yields to the detail signal off(x)
at the resolution 2 ‘. It is characterized by the set of inner products
Dzif = (( f(u)
1 iczi(u - 2-‘4
,),l,z.
(21)
Dz,f is called the discrete detail signal at the resolution
2 J. It contains the difference of information between
A:,- 1f and A$ f. As we did in (12), we can prove that each
of these inner products is equal to the convolution off(x)
with &,( -x) evaluated at 2-/n
(f(u),
$?,(U - 2-52)) = (f(u) * ic?,(-u))(2-'n).
(22)
Equations (21) and (22) show that the discrete detail signal at the resolution 2’ is equal to a uniform sampling of
(f(u) * rl/?,( -u)) (x) at the rate 2’
&if = ((f(u) * h-4)
(2-/")),,Ez.
The wavelet $(x) can be viewed as a bandpass filter
whose frequency bands are approximatively
equal to
vJv(x
1
1
0.5
0
-0.5
:+
0.4
1
0.2
0
1
t
Fig. 4. (a) Wavelet $(I) associated to the scaling function of Fig. 1. (b)
Modulus of the Fourier transform of I/J(X). A wavelet is a band-pass
filter.
[ -277, -x] U [K, 2~1. Hence, the detail signal, D?,f
describes f(x) in the frequency bands [ -2 -‘+ ’x,
-2-‘7r]
u [2-JT, 2-‘+‘7r].
We can prove by induction that for any J > 0, the original discrete signal A:f measured at the resolution 1 is
represented by
(AP Jf,
(Dz~fIp~~j<-l).
(23)
This set of discrete signals is called an orthogonal wavelet
representation, and consists of the reference signal at a
coarse resolution A: if and the detail signals at the resolutions 2’ for -J I j I - 1. It can be interpreted as a
decomposition of the original signal in an orthonormal
wavelet basis or as a decomposition of the signal in a set
of independent frequency channels as in Marr’s human
vision model [30]. The independence is due to the orthogonality of the wavelet functions.
It is difficult to give a precise interpretation of the model
in terms of a frequency decomposition because the overlap of the frequency channels. However, we can control
this overlap thanks to the orthogonality of our decomposition functions. That is why the tools of functional analysis give a better understanding of this decomposition. If
we ignore the overlapping spectral supports, the interpretation in the frequency domain provides an intuitive approach to the model. In analogy with the Laplacian pyramid data structure, Ai rfprovides the top-level Gaussian
pyramid data, and the D?,,f data provide the successive
681
MALLAT: THEORY FOR MULTIRESOLUTION SIGNAL DECOMPOSITION
Laplacian pyramid levels. Unlike the Laplacian pyramid,
however, there is no oversampling, and the individual
coefficients in the set of data are independent.
B. Implementation of an Orthogonal Wavelet
Representation
In this section, we describe a pyramidal algorithm to
compute the wavelet representation. With the same
derivation steps as in Section II-B, we show that D2, f can
be calculated by convolving A;‘,+, f with a discrete filter
G whose form we will characterize.
For any n E Z, the function &, (X - 2-’ n ) is a member
of 02, c Vz,- I. In the same manner as (13)) this function
can be expanded in an orthonormal basis of V, I + I
I&,(X - 2-/n)
=
keep one sample out of Iwo
l-7-j
* &,*l(X
,t”..
($,,(u
- 2-/n),
qlh,.I(U - 2p’-‘k))
- 2p’eq.
(24)
As we did in (14), by changing variables in the inner
product integral we can prove that
2-‘-I(
&!(u
- 2-jn),
&,+I(u
- 2-‘-‘k))
(25)
Hence, by computing the inner product off(x)
functions of both sides of (24), we obtain
= ,jfm (bl(u>,
. (f(u),
uses a symmetry with respect to the first and the last sample as in Section II-B.
Equation (19) of Theorem 3 implies that the impulse
response of the filter G is related to the impulse response
of the filter H by
+(u - (k - 2n)))
&,+,(u - 2-‘-Q)).
g(n) = (4&l(u), 4(u - n)),
(26)
(27)
and G be the symmetric filter with impulse response g (n )
= g ( -n ). We show in Appendix D that the transfer function of this filter is the function G(w) defined in Theorem
3, equation (19). Inserting (27) to (26) yields
bh,(u - 2:'n))
= jY,"(Zn
(-l)‘-”
=
h(1 - n).
(29)
This equation is provided in Appendix D. G is the mirror
filter of H, and is a high-pass filter. In signal processing,
G and H are called quudrature mirror filters [lo]. Equation (28) can be interpreted as a high-pass filtering of the
discrete signal A;‘, +of.
If the original signal has N samples, then the discrete
signals D2, f and A:, f have 2’ N samples each. Thus, the
wavelet representation
with the
Let G be the discrete filter with impulse response
(f(u),
X
Fig. 5. Decomposition of a discrete approximation A$, , ,f into an approximation at a coarser resolution A$ j’and the signal detail LLf. By repeating in cascade this algorithm for -1 2 j 2 -J, we compute the
wavelet representation of a signal A‘:fon J resolution levels.
g(n)
2-‘-l
convolve with IMer
- k) (f(u),c#Q,+,(u - 2-‘-‘/c)).
(28)
Equation (28) shows that we can compute the detail signal
D,, f by convolving At,- ,f with the filter G and retaining
every other sample of the output. The orthogonal wavelet
representation of a discrete signal Aff can therefore be
computed by successively decomposing At,, ,f into At, f
and Dz,f for -J 5 j I - 1. This algorithm is illustrated
by the block diagram shown in Fig. 5.
In practice, the signal A;‘f has only a finite number of
samples. One method of handling the border problems
has the same total number of samples as the original approximated signal A: f. This occurs because the representation is orthogonal. Fig. 6(b) gives the wavelet representation of the signal Ayfdecomposed in Fig. 2. The energy
of the samples of D,, f gives a measure of the irregularity
of the signal at the resolution 2’+‘. Whenever AI, f (x)
and A2,+ If(x) are significantly different, the signal detail
has a high amplitude. In Fig. 6, this behavior is observed
in the textured area between the abscissa coordinates of
60 and 80.
C. Signal Reconstruction from an Orthogonal Wavelet
Representation
We have seen that the wavelet representation is complete. We now show that the original discrete signal can
also be reconstructed with a pyramid transform. Since
Oz, is the orthogonal complement of V, I in VI,+ I, ( fi
&)(x - 2-jn), J2-l Ic/?,(x - 2Pin)),,EZ is an orthonorma1 basis of VI ,+ I. For any n > 0, the function &,+ I (X
- 2-j- ’n ) can thus be decomposed in this basis
&,+I(x
- 2-‘-In)
= 2-‘,:<-
(&,(u
- 2-/k),
&,-I(U
- 2:‘-‘n))
. $2,(x - 2-‘/k)
+ 2-j j!Zy,
($?,(u
- 2-/k),
* y&,(x - 2-/i?).
&,+I(u
- 2:‘-‘n))
(30)
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-+Ajl;fs$+,f
IT?I
+
put one zero between each sample
convolve wlh tiller
a:
: multiplication by
1.
19X9
X
2
Fig. 7. Reconstruction of a discrete approximation A!, I ffrom an approximation at a coarser resolution &,f and the signal detail D,, f. By repeating in cascade this algorithm for -.I I j 5 - 1, we reconstruct A:f
from its wavelet representation.
yields
4*,+1(U - 2-‘-w
(f(u),
(a)
= 2 ,g
Az-sfd
m
+ 2 &+
h(n
-
2k) (f(u),+&
- 2k)(f(u),
-
$*,(u
2:+))
-
2-JJk)).
'
Dpf_
(32)
D2-6.’
..‘:_,,
D2-:fe
Dpf
..,;
7’
.:,
.
: ,‘. ,:
.:::
‘..
;
Y..,..
.._.
Fig. 6. (a) Multiresolution continuous approximationsAz,f(X).
(b) Wavelet representation of the signal A,f(x).
The dots give the amplitude of
the inner products ( f(u),
$?, (u - 2 -‘n) ) of each detail signal Dz,f
depending upon 2-‘n. The detail signals samples have a high amplitude
when the approximationsAzrf(X)
andA,,+,f(*)
shown in (a) are locally
different. The top graph gives the inner products ( f( u ). 4 J (u - 2 ‘II) )
of the coarse discrete approximation A$,f.
By computing the inner product of each side of equation
(30) with the functionf(x),
we have
This equation shows that A$, + I f can be reconstructed by
putting zeros between each sample of A&f and &, f and
convolving the resulting signals with the filters H and G,
respectively. A quite similar process can be found in the
reconstruction algorithm of Burt and Adelson from their
Laplacian pyramid [5].
The block diagram shown in Fig. 7 illustrates this algorithm. The original discrete signal A;‘fat the resolution
1 is reconstructed by repeating this procedure for -1 5
j < 0. From the discrete approximation Aff, we can recover the continuous approximation A, f(x) with equation (10). Fig. 8(a) is a reconstruction of the signal A, f(x)
from the wavelet representation given in Fig. 6(b). By
comparing this reconstruction with the original signal
shown in Fig. 8(b), we can appreciate the quality of the
reconstruction. The low and high frequencies of the signal
are reconstructed well, illustrating the numerical stability
of the decomposition and reconstruction processes.
IV.
EXTENSION OF THE ORTHOGONAL
REPRESENTATION
(f(u),
q!J*,+,(u = 2-j&y
(qb*,(u
* (f(u),
+ 2-j j<,
2-'-51))
($,,(u
* (f(u),
h(u
- 2%c),
-
-
- 2-'%I))
2-w
- 2-'k),
$r,(u
4*,+1(u
c#Q,+,(u - 2-'-'4)
2-q).
(31)
Inserting (14) and (25) in this expression and using the
filters H and G, respectively, defined by (15) and (27)
WAVELET
TO IMAGES
The wavelet model can be easily generalized to any dimension y1 > 0 [33]. In this section, we study the twodimensional case for image processing applications. The
signal is now a finite energy functionf(x,
y) E L*(R’).
A multiresolution approximation of L2 (R*) is a sequence
of subspaces of L’ (R’) which satisfies a straightforward
two-dimensional extension of the properties (2) to (8). Let
(Vz, )jez be such a multiresolution
approximation of
L* ( R2). The approximation of a signal f(x, y ) at a resolution 2 J is equal to its orthogonal projection on the vector space V,,. Theorem 1 is still valid in two dimensions,
and one can show that there exists a unique scaling function @(x, y) whose dilation and translation given an or-
MALLAT.
THEORY
FOR MULTIRESOLUTION
SIGNAL
DECOMPOSITION
Fig. 9. Approximations
l/8
of an image at the resolutions 1, l/2,
(j = 0, -I, -2, -3).
l/4,
and
The approximation of a signal f(x, y) at a resolution 2 ’
is therefore characterized by the set of inner products
A&f =
(( f(x,
(b)
Fig. 8. (a) Original sIgnal A,,~(.I ) approximated at the resolution I. (b)
Reconstruction of A, f(.r) from the wavelet representation shown in Fig.
6(b). Bj comparing both figures. we can appreciate the quality of the
reconstruction.
thonormal basis of each space I’?,, . Let @ ‘2,(x, y) = 2”
@ (2”x, 2’~). The family of functions
pb(x
- 2-,/n, \’ - 2-‘m))(n,,,,)E72
forms an orthonormal basis of V,,. The factor 2-’ normalizes each function in the L2(R2) norm. The function
+ (x, .v) is unique with respect to a particular multiresolution approximation of L’ (R 2 ).
W e next describe the particular case of separable multiresolution approximations of L’( R2) studied by Meyer
[35]. For such multiresolution approximations, each vector space If,, can be decomposed as a tensor product of
two identical subspaces of L2 (R )
v,, = v:, 0 v:,.
The sequence of vector spaces ( V2,),Ez forms a multiresolution approximation of L*( R2) if and only if
(VL),,EZ is a multiresolution approximation of L* (R ). One
can then easily show that the scaling function + (x, y ) can
be written as
Y>? 42,(x - 2w
(2p%‘b
- 2-h
= phh(~
))i,z,,,,irZl
@(x, y) = 4,(x> G(y),
*‘(x,
**(x3 .v> = $(x) 4,(y),
vv) = $(x> $(Y)
are such that
- 2-/n, y - 2:/m).
2:‘*:,(x
- 2:ln, y - 2:jrn),
2-9:,(x
- 2-/n, ?’ - 2-‘rn))(,~,,,~)Fz’
(34)
is an orthonormal basis of 02, and
(2-‘9j,(x
- 2-‘n, y - 2-/m),
2-Q!;, (x - 2-/n, y - 2-/m),
?’ - 2-‘mj)(,,,,,,,Ez2
- 2-/n) &I(?
- 2-w
Let us suppose that the camera measures an approximation of the irradiance of a scene at the resolution 1. Let
Affbe the resulting image and N be the number of pixels.
One can easily show that, for j < 0, a discrete image approximation At,f has 2 JN pixels. Border problems are
handled by supposing that the original image is symmetric
with respect to the horizontal and vertical borders. Fig. 9
gives the discrete approximations of an image at the resolutions 1, l/2, l/4, and l/8.
As in the one-dimensional case, the detail signal at the
resolution 2’ is equal to the orthogonal projection of the
signal on the orthogonal complement of V2, in V2, + I. Let
01, be this orthogonal complement. The following theorem gives a simple extension of Theorem 3, and states
that we can build an orthonormal basis of 02, by scaling
and translating three wavelets functions, * ’(x, y), q2 (x,
y), and *j(x,
y).
Theorem 4: Let ( V2,)jGz be a separable multiresolution
approximation of L2(R2). Let Q(.x, y) = C#I(X) $(y) be
the associated two-dimensional
scaling function. Let
\k (x) be the one-dimensional wavelet associated with the
scaling function C#I(x). Then, the three “wavelets”
(2-‘Ql,(x
where C#I
(x) is the one-dimensional scaling function of the
multiresolution approximation ( V:,)jEz. W ith a separable
multiresolution approximation, extra importance is given
to the horizontal and vertical directions in the image. For
many types of images, such as those from man-made environments, this emphasis is appropriate. The orthogonal
basis of V2, is then given by
h(Y
- 2-‘m))(,l,,,l,tz1.
(33)
2-)*G,(x
- 2-h,
Y - 2e’m))(,,,,n,Ez3
is an orthonormal basis of L’( R’).
(35)
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Appendix E gives a proof of this theorem. The difference of information between A$ + I f and A$ f is equal to
the orthonormal projection of f(x) on 02,, and is characterized by the inner products of f(x) with each vector
of an orthonormal basis of O,, . Theorem 4 says that this
difference of information is given by the three detail images
D;,f
= (( j-(x, Y), *:,cx
- 2-k
Y - 2-“m) ))ln,m)FZ?:
(36)
D&f = ((j-(x, y), *:,(X - 2-k Y - 2-Jm)))i,,,,n)FZ?
D;,f
= (( f(x,
Y), ‘P;,(x
- 2-k
Y - 2-44
(37)
))ir~,,,I~EZ~.
(38)
Just as for one-dimensional signals, one can show that in
two dimensions the inner products which define A’:,f,
0: ,f, 0; ,f, and D:, f are equal to a uniform sampling of
two-dimensional convolution products. Since the three
wavelets \k,(x, y), \k>(x, y), and \k3(x, y) are given by
separable products of the functions 4 and $, these convolutions can be written
A$,f =
((fb
Y> * +21(-x)
d)*d-Y))
(b)
Fig. 10. (a) Decomposition of the frequency support of the image A!, 1f
into A’,‘,fand the detail images LIl f. The image A’,‘,fcorresponds to the
lower horizontal and vertical frequencies of A,, ,_ Di,fgives the vertical
high frequencies and horizontal low frequencies. L2-, J‘the horizontal high
frequencies and vertical low frequencies and Dl,fthe
high frequencies
in both horizontal and vertical directions. (b) Disposition of the Di,f
and A$ , f’ images of the image wavelet representations shown in this
article.
F’n, 2-‘m))(,l.,n)Ez1
(39)
D;,f
=
2-‘n, 2-Jm))(,1.,r,,Ez1
(40)
Dfif
=
((fb
(a)
~9 *
Ir/2,( -4
h(
-Y)>
(2-k
2pJm)),,l,,n)Ez2
(41)
D;,f
=
The expressions (39) through (42) show that in two dimensions, A$, f and the Di, f are computed with separable
filtering of the signal along the abscissa and ordinate.
The wavelet decomposition can thus be interpreted as a
signal decomposition in a set of independent, spatially
oriented frequency channels. Let us suppose that C#I
(x)
and $(x) are, respectively, a perfect low-pass and a perfect bandpass filter. Fig. 10(a) shows in the frequency domain how the image A$+ I f is decomposed into A,“, f,
04, f, D:, f, and D:, f. The image A:, f corresponds to the
lowest frequencies, D:, f gives the vertical high frequencies (horizontal edges), Ds, f the horizontal high frequencies (vertical edges) and D:, f the high frequencies in both
directions (the comers). This is illustrated by the decomposition of a white square on a black background explained in Fig. 11(b). The arrangement of the Di,f im-
Cc)
Fig. I I. (a) Original image. (b) Wavelet representation on three resolution
levels. The black, grey. and white pixels correspond respectively to negative, zero. and positive wavelet coefficients. The disposition of the detatl images is explained in Fig. IO(b). (c) These images show the absolute value of the wavelet coefficients for each detail images Di, shown
in (b). Black and white pixels correspond respectively to zero and high
amplitude coefficients. The amplitude is high along the edges of the
square for each orientation.
685
MALLAT: THEORY FOR MULTIRESOLUTIONSIGNAL DECOMPOSITION
ages is shown in Fig. 10(b). The black, grey, and white
pixels respectively correspond to negative, zero, and positive coefficients. Fig. 1 l(c) shows the absolute value of
the detail signal samples. The black pixels correspond to
zero whereas the white ones have a high positive value.
As expected, the detail signal samples have a high amplitude on the horizontal edges, the vertical edges and the
comers of the square.
For any J > 0, an image A;(fis completely represented
by the 3J + 1 discrete images
m
:
keeo one row
11121:
This set of images is called an orthogonal wavelet representation in two dimensions. The image Af-,f is the
coarse approximation at the resolution 2PJ and the D:,f
images give the detail signals for different orientations and
resolutions. If the original image has N pixels. each image
Af?‘,f, DJ,f, D<,f, Dz,fhas 2/N pixels ( j < 0). The total
number of pixels in this new representation is equal to the
number of pixels of the original image, so we do not increase the volume of data. Once again, this occurs due to
the orthogonality of the representation. In a correlated
multiresolution representation such as the Laplacian pyramid. the total number of pixels representing the signal is
increased by a factor of 2 in one dimension and of 4/3 in
two dimensions.
A. Decomposition and Reconstruction Algorithms in
TN)ODimerzsiorzs
In two dimensions, the wavelet representation can be
computed with a pyramidal algorithm similar to the onedimensional algorithm described in Section III-B. The
two-dimensional wavelet transform that we describe can
be seen as a one-dimensional wavelet transform along the
s and y axes. By repeating the analysis described in Section III-B, we can show that a two-dimensional wavelet
transform can be computed with a separable extension of
the one-dimensional decomposition algorithm. At each
step we decompose A&+ of into A;,f, D$, f, D$, f, and
02) f. This algorithm is illustrated by a block diagram in
Fig. 12. We first convolve the rows of A;, , , f with a onedimensional filter, retain every other row, convolve the
columns of the resulting signals with another one-dimensional filter and retain every other column. The filters used
in this decomposition are the quadrature mirror filters fi
and e described in Sections II-B and III-B.
The structure of application of the filters for computing
A:, . 04, , Di, , and D$, is given in Fig. 12. We compute
the wavelet transform of an image A: f by repeating this
process for - 1 2 j 2 -J. This corresponds to a separable conjugate mirror filter decomposition [44].
Fig. 14(b) shows the wavelet representation of a natural
scene image decomposed on 3 resolution levels. The pattern of arrangement of the detail images is as explained
in Fig. 10(b). Fig. 14(c) gives the absolute value of the
wavelet coefficients of each detail image. The wavelet
convolve (rows or columns) wilh the fuller X
keep one column ouL of two
I?-I’
out
of
two
Fig. 12. Decomposition of an image A:,. f into A$,f. Dlflf. Di,f. and
D:,f: This algorithm is based on one-dimensional convolutions of the
rows and columns of A’,‘.. fwith the one dimensional quadrature mirror
filters fi and G.
coefficients have a high amplitude around the images
edges and in the textured areas within a given spatial orientation.
The one-dimensional
reconstruction
algorithm
described in Section III-C can also be extended to two dimensions. At each step, the image A$+ of is reconstructed
from A;‘, f, Di,,f, Dz, f, and 02, f. This algorithm is illustrated by a block diagram in Fig. 13. Between each column of the images A$ f, D:, f, Ds, f, and Dz, f, we add a
column of zeros, convolve the rows with a one dimensional filter, add a row of zeros between each row of the
resulting image, and convolve the columns with another
one-dimensional filter. The filters used in the reconstruction are the quadrature mirror filters Hand G described in
Sections II-B and III-B. The image Ayf is reconstructed
from its wavelet transform by repeating this process for
-J i ,j i - 1. Fig. 14(d) shows the reconstruction of
the original image from its wavelet representation. If we
use floating-point precision for the discrete signals in the
wavelet representation, the reconstruction is of excellent
quality. Reconstruction errors are more thoroughly discussed in the next section.
V. APPLICATIONS OF THE ORTHOGONAL WAVELET
REPRESENTATION
A. Compact Coding
of
Wavelet Image Representations
To compute an exact reconstruction of the original image, we must store the pixel values of each detail image
with infinite precision. However, for practical applications, we can allow errors to occur as long as the relevant
information is not destroyed for a human observer. In this
section, we show how to use the sensitivity of the human
visual system as well as the statistical properties of the
image to optimize the coding by the wavelet representation. The conjugate mirror filters which implement the
wavelet decomposition have also been studied by Woods
[44] and Adelson et al. [l] for image coding.
Let
(A$,f,
(D;if LJs,,s -17
(D:,f >pJs,s -13
(D:, f ) -.,<, 5 ~I ) be the wavelet representation of an
image A;‘f.
Let (E-J,
(+Js,+l~
($-Js,+l?
( E: ) -J <, 5 ~, ) be the mean square errors introduced when
coding each image component of the wavelet representa-
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Eo = 22Jt-J + c
k=I
A$+I f
convoive (row or column) with filter
;,
gj
X
put one row of zeros between each row
put one column of zeros between each column
multiply by 4
Fim
C’ 13. Reconstruction of an image A$ f from A$f, Di,f; Df,f. and
0: jY The row and columns of these images are convolved with the one
dimensional quadrature mirror filtera H and C.
(a)
(b)
(4
Fig. 14. (a) Original image. (b) Wavelet representation on three resolution
levels. The arrangement of the detail images ia explained in Fig. IO(b).
Cc) These images show the absolute value of the wavelet coefficients for
each detail images Di, shown in (b). The amplitude is high along the
edges and the textured area for each orientation. (d) Reconstruction of
the original image from the wavelet representation given in (b).
tion. Let cObe the mean square error of the image reconstructed from the coded wavelet representation. Since the
wavelet representation is orthogonal in L’(R’),
one can
VOL.
II.
NO
7. JULY
lY8Y
-1
c 2-y.
,,=J
The factors 2-2’ are due to the normalization factor 2-j
which appears in (33) and (34). Psychological experiments on human visual sensitivity show that the visible
distortion on the reconstructed image will not only depend
on the total mean square error co, but also on the distribution of this error between the different detail images
Di, f. The contrast sensitivity function of the visual SYStern [6] shows that the perception of a contrast distortion
in the image depends upon the frequency components of
the modified contrast. Visual sensitivity also depends upon
the orientation of the stimulus. The results of Campbell
and Kulikowski [7] show that the human visual system
has a maximum sensitivity when the contrast is horizontal
or vertical. When the contrast is tilted at 45’) the sensitivity is minimum. A two-dimensional wavelet representation corresponds to a decomposition of the image into
independent frequency bands and three spatial orientations. Each detail image Di, gives the image contrast in
a given frequency range and along a particular orientation. It is therefore possible to adapt the coding error of
each detail image to the sensitivity of human perception
for the corresponding frequency band and spatial orientation selectivity. The more sensitive the human visual
system, the less coding error we want to introduce in the
detail image 05, f. Watson [43] has made a particularly
detailed study of subband image coding adapted to human
visual perception.
Given an allocation of the coding error between the different resolutions and orientations of the wavelet representation, we must then code each detail image with a
minimum number of bytes. In order to optimize the coding, one can use the statistical properties of the wavelet
coefficients for each resolution and orientation. Natural
images are special kinds of two-dimensional signals. This
shows up clearly when one looks at the histogram of the
detail images D$ If. Since the pixels of these detail images
are the decomposition coefficients of the original image
in an orthonormal family, they are not correlated. The
histogram of the detail images could therefore have any
distribution. Yet in practice, for all resolutions and orientations, these histograms are symmetrical peaks centered in zero. Natural images must therefore belong to a
particular subset of L’( R’). The modeling of this subset
is a well known problem in image processing [26]. The
wavelet orthonormal bases are potentially helpful for this
purpose. Indeed, the statistical properties of an image decomposition in a wavelet orthonormal basis look simpler
than the statistical properties of the original image. Moreover, the orthogonality of the wavelet functions can simplify the mathematical analysis of the problem.
We have found experimentally that the detail image histograms can be modeled with the following family of histograms:
/l(u) = Ke-‘l”l/,)J.
(44)
MALLAT:
THEORY
FOR MULTIRESOLUTION
SIGNAL
The parameter 0 modifies the decreasing rate of the peak
and CYmodels the variance. This model was built by studing the histograms of seven different images decomposed
on four resolution levels each. Our goal was only to define
a qualitative histogram model. The constant K is adjusted
in order to have ST, h(u) du = N where N is the total
number of pixels of the given detail image. By changing
variables in the integral, one can derive that
K=- NP,
where
I’(r)
=
m
s e-Q-’
0
687
DECOMPOSITION
du.
F-‘(x)
t
1.5
1
0.5
:_;
(45)
OA
0.2
0.4
0
Fig. 15. Graph of the function F-‘(x)
The coefficients CYand @ of the histogram model can be
computed by measuring the first and second moment of
the detail image histogram:
X
characterized by (49)
h(u) r
03
ml =
-m lulh(u)du
and m2 =
s
m u2h(u) du.
c --co
15000
(46)
By inserting (44) of the histogram model and changing
variables in these two integrals, we obtain
ml =
and m 2- -2Koi71’
(47)
10000
5000
0
P
Thus.
(a)
where
(YIP
(48)
0
1
m217 P
\
0
,
(49)
-10
-5
0
,,,,,,,,j
5
10
u
(b)
Nr
;
0
The function F-‘(x) is shown in Fig. 15. Fig. 16(a) gives
a typical example of a detail image histogram obtained
from the wavelet representation of a real image. Fig. 16(b)
is the graph of the model derived from (44).
The a priori knowledge of the detail signal’s statistical
distribution given by the histogram model (44) can be used
to optimize the coding of these signals. W e have developed such a procedure [27] using Max’s algorithm [32] to
minimize the quantization noise on the wavelet representation. Predictive coding procedures are also effective for
this purpose. Results show that one can code an image
using such a representation with less than 1.5 bits per pixel
with few visible distortions [ 11, [27], [44].
B. Texture Discrimination and Fractal Analysis
W e now describe the application of the wavelet orthogonal representation to texture discrimination and fractal
Fig. 16. (a) Typical example of a detail image histogram h(u). (b) Modeling of h ( U) obtained from equation (44). The parameters oi and 0 have
been computed from the first two moments of the original histogram (01
= 1.39andP = 1.14).
analysis. Using psychophysics, Julesz [21] has developed
a texture discrimination theory based on the decomposition of textures into basic primitives called textons. These
textons are spatially local; they have a particular spatial
orientation and narrow frequency tuning. The wavelet
representation can also be interpreted as a texton decomposition where each texton is equal to a particular function of the wavelet orthonormal basis. Indeed, these functions have all the discriminative abilities required by the
Julesz theory. In the decomposition studied in this article,
we have used only three orientation tunings. However,
one can build a wavelet representation having as many
orientation tunings as desired by using non-separable
wavelet orthonormal bases [33].
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Fig. 17(a) shows three textures synthesized by Beck.
Humans cannot preattentively discriminate the middle
from the right texture but can separate the left texture from
the others. In this example, human discrimination is based
mainly on the orientation of these textures as their frequency content is very similar. With a first-order statistical analysis of the wavelet representation shown in Fig.
17(b), we can also discriminate the left texture but not the
two others. This example illustrates the ability of our representation to differentiate textures on orientation criteria.
This is of course only one aspect of the problem, and a
more sophisticated statistical analysis is needed for modeling textures [ 131. Although several psychophysical
studies have shown the importance of a signal decomposition in several frequency channels [4], [ 151, there still
is no statistical model to combine the information provided by the different channels. From this point of view,
the wavelet mathematical model might be helpful to transpose some of the tools currently used in functional analysis to characterize the local regularity of functions [25].
Mandelbrot [29] has shown that certain natural textures
can be modeled with Brownian fractal noise. Brownian
fractal noise F(x) is a random process whose local differences
(F(x)
ANALYSIS
AND
MACHINE
INTELLIGENCE.
VOL.
II.
NO. 7. JULY
1%‘)
(a)
(b)
Fig. 17. (a) .I. Beck textures: only the left texture 15 preattcntively discrlminable by a human observer. (h) These Images show the alxolutc value
of the wavelet coefficients of image (a). computed on three resolution
levels. The left texture can be discrimmated with a first-order statistical
analysis of the detail signals amplitude. The two other textures can not
be dlscriminatcd with such a technic.
- F(x + Ax) /
I/Ax/IH
has a probability distribution function g(x) which
Gaussian. Such a random process is self-similar, i.e..
tlr > 0, F(x)
and
rHF(rx)
is
are statistically identical.
(a)
Hence, a realization of F(x) looks similar at any scale
and for any resolution. Fractals do not provide a general
model which can be used for the analysis of any kind of
texture, but Pentland [39] has shown that for a fractal texture, the psychophysical perception of roughness can be
quantified with the fractal dimension.
Fig. 18(a) shows a realization of a fractal noise which
looks like a cloud. Its fractal dimension is 2.5. Fig. 18(b)
gives the wavelet representation of this fractal. As expected, the detail signals are similar at all resolutions. The
image Ai-if gives the local dc component of the original
fractal image. For a cloud, this would correspond to the
local differences of illuminations.
Let US show that the fractal dimension can be computed
from the wavelet representation. We give the proof for
one-dimensional fractal noise, but the result can be easily
extended to two dimensions. The power spectrum of fractal noise is given by [29]
P(w)
(b)
Fig. 18. (a) Brownian fractal image. (b) Wavelet representation on three
resolution levels of image (a). A\ expected. the detail signals are similar
at all resolutions.
interpreted in the classical sense. Flandrin [ 121 has shown
how to define precisely this power spectrum formula with
a time-frequency analysis. We saw in equation (22) that
the detail signals Dz,fare obtained by filtering the signal
with &, ( -x) and sampling the output. The power spectrum of the fractal filtered by &, ( -x) is given by
= kcC2Hp’.
(50)
The fractal dimension is related to the exponent H by
D=T+l-H
(51)
where T is the topological dimension of the space in which
x varies ( for images T = 2 ). Since Brownian fractal noise
is not a stationary process, this power spectrum cannot be
P2#(W) = P(w) I5(2-‘w)~?.
(52)
After sampling at a rate 2’, the power spectrum of the
discrete detail signal becomes [37]
P;,(w)
= 2J ,;<m P,,(w
+ 2/2kr).
(53)
VALI.&T
THEORY FOR MULTIRESOLUTION SIGNAL DECOMPOSITION
Let ai, be the energy of the detail signal Dz, f
*217r
2-J
o;, = -2T -2,n mJJ) h.
\’
(54)
By inserting (52) and (53) into (54) and changing variables in this integral, we obtain that
2 = 2’Ho5 / 1 1.
02,
(55)
For a fractal, the ratio ai,/ai,&r
is therefore constant.
From the wavelet representation of a Brownian fractal, we
compute H from equation (55) and derive the fractal dimension D with (51). This result can easily be extended
to two dimensions in order to compute the fractal dimension of fractal images. Analogously, we compute the ratios of the energy of the detail images within each direction, and derive the value of the coefficient H. A similar
algorithm has been proposed by Heeger and Pentland for
analyzing fractals with Gabor functions [ 191. For the fractal shown in Fig. 18(a), we calculated the ratios of the
energy of the detail images within each orientation for the
resolutions l/2. l/4, and l/8. We recovered the fractal
dimension of this image from each of these ratios with a
3 percent maximum error.
Much research work has recently concentrated on the
analysis of fractals with the wavelet transform [2]. This
topic is promising because multiscale decompositions,
such as the wavelet transform, are well adapted to evaluate the self-similarity of a signal and its fractal properties.
VI. CONCLUSION
This article has described a mathematical model for the
computation and interpretation of the concept of a multiresolution representation. We explained how to extract the
difference of information between successive resolutions
and thus define a new (complete) representation called the
wavelet representation. This representation is computed
by decomposing the original signal using a wavelet orthonormal basis, and can be interpreted as a decomposition using a set of independent frequency channels having
a spatial orientation tuning. A wavelet representation lies
between the spatial and Fourier domains. There is no redundant information because the wavelet functions are orthogonal. The computation is efficient due to the existence of a pyramidal algorithm based on convolutions with
quadrature mirror filters. The original signal can be reconstructed from the wavelet decomposition with a similar algorithm.
We discussed the application of the wavelet representation to data compression in image coding. We showed
that an orthogonal wavelet transform provides interesting
insight on the statistical properties of images. The orientation selectivity of this representation is useful for many
applications. We reviewed in particular the texture discrimination problem. A wavelet transform is particularly
well-suited to analyze the fractal properties of images.
Specifically, we showed how to compute the fractal dimension of a Brownian fractal from its wavelet represen-
689
tation. In this article, we emphasized the computer vision
applications, but this representation can also be helpful
for pattern recognition in other domains. Grossmann and
Kronland-Martinet
[23] are currently working on speech
recognition applications, and Morlet [ 141 studies seismic
signal analysis. The wavelet orthonormal bases are also
studied in both pure and applied mathematics [20], [25],
and have found applications in Quantum Mechanics with
the work of Paul [38] and Federbush [ 111.
APPENDIX A
AN
EXAMPLE
MULTIRESOLUTION
APPROXIMATION
In this appendix, we describe a class of multiresolution
approximations of L’(R) studied by Lemarie [24] and
Battle [3]. We explain how to compute the corresponding
scaling functions 4 (x), wavelets G(x), and quadrature
filters H. These multiresolution approximations are built
from polynomial splines of order 2p + 1. The vector space
V, is the vector space of all functions of L’( R) which are
p times continuously differentiable and equal to a polynomial of order 2p -t 1 on each interval [k, k + 11, for
any k E Z. The other vector spaces V,, are derived from
V, with property (3). Lemarie has shown that the scaling
function associated with such a multiresolution approximation can be written
.
4(w)
where
= d&j
n = 2 + 2p,
(56)
and where the function C,,(w) is given by
C,,(w)
=
1
E
h=-w (w + 2kr)“’
(57)
We can compute a closed form of C,, (0) by calculating
the derivative of order y1 - 2 of the equation
C*(w) =
l
4 sin* (w/2)’
Theorem 2 says that a(w) is related to the transfer function H( w ) of a quadrature mirror filter by
$(2w)
= H(w) $(w).
From (56) we obtain
H(w)
=
C*,,(w)
\i
2?17c2,,(2w)’
(58)
The Fourier transform of the corresponding orthonormal
wavelet can be derived from the property (19) of Theorem
3
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= 5 + 30 ( cos 2-)‘+
II.
NO
7. JULY
1989
TABLE1
The wavelet $(x) defined by (59) decreases exponentially.
The scaling function shown in Fig. 1 was obtained with
p = 1, and thus n = 4. It corresponds to a multiresolution
approximation built from cubic splines. Let
N,(o)
VOL.
h(n)
3O(sin~)?(cos~j
0.006
0.006
and
-0.003
N*(a)
= 2(sini)i(iorq)?
+ 7O(cOs~)
-0.002
By inserting (62) in (60), we obtain
/ +CX?
M(w) =
The function Cs( w) is given by
c,(w)
= N,(w)
+ h(w)
105 sin i
c
Y.
One can show that the continuity of the isomorphism Z
implies that there exists two constants C, and C2 such that
For this multiresolution approximation based on cubic
splines, the functions 6 (w ) and 5 (w) are computed from
(56) and (59) with n = 4. The transfer function H(w) of
the quadrature mirror filter is given by equation (58). Table I gives the first 12 coefficients of the impulse response
This filter is symmetrical. The impulse re(h(n))n,z.
sponse of the mirror filter G is obtained with (29).
APPENDIX
B
1
This appendix gives the main steps of the proof of
Theorem 1. More details can be found in [28]. We prove
Theorem 1 forj = 0. The result can be extended for any
j E Z using the property (3). From the properties (5) and
(6) of the isomorphism Z from VI onto Z*(Z ), one can
prove that there exists a function g (x) such that ( g (X k))kEz is a basis of VI. We are looking for a function
4(x) E V, suchthat (4(x - k))kEZ is an orthonormal
basis of V, Let 4 (w ) be the Fourier transform of $ (x).
With the Poisson formula, we can show that the family of
functions ($(x - k))kEZ is orthonormal if and only if
I/*
C, I
x
2ka)(*
= 1.
(60)
Since 4 (x) E V,, it can be decomposed in the basis ( g (X
- k)),,,:
+-x)kEZ
E z2w
such that
$(x)
= k:
akg(x
I
c*.
!
Hence, (63) is defined for any
simple to prove that equations
Fourier transform of a function
k))kcz is an orthonormal family
PROOF OF THEOREM
,g, I$(w +
Ig(w + 2ka)l*
w E R. Conversely, it is
(62) and (63) define the
4 (x) such that (4(x that generates VI.
C
APPENDIX
PROOF
OF THEOREM
2
This appendix gives the main steps of the proof of
Theorem 2. More details can be found in [28]. Let us first
prove property (17a). Since $2m1(~) E V-, C V,, it can
be decomposed in the orthogonal basis (4 (x - k) ) kc~
42-'(x)
=
,g,
(+2-W
4(u
-
4)
+(u
The Fourier transform of this equation yields
6P-4
= H(w)
6(w)
From (64) we obtain 1H(0) 1 = 1. Since the asymptotic
decay of 4 (x) at infinity satisfies
)4(x))
= w-x-2)
we can also derive that
with
M(w)
= ,Iz’,
(64)
where H( w ) is the Fourier series defined by (17). One
can show that the property (7) of a multiresolution approximation implies that any scaling function satisfies
The Fourier transform of (61) can be written
= M(w) g(w)
k).
- k).
VW
$(a)
-
Coke’““.
(62)
h(n) = (q!-‘(u), 4(u - rl)) = 0(n-‘)
at infinity.
691
MALLAT. THEORY FOR MULTIRESOLUTlONSIGNAL DECOMPOSITION
APPENDIX D
PROOF OF THEOREM 3
Let us now prove property ( 17b). We saw in Appendix
A that a scaling function must satisfy
,g., I$(
w + 2k7r)12 = 1.
Since H(w)
is 2~ periodic,
(H(w)(2
(66)
(64) and (66) yield
+ /H((w
+ 7r)f
= 1.
This appendix gives the main steps of the proof of
Theorem 3. More details can be found in [28]. This theorem is proved for j = - 1. We are looking for a function
$(x) E L’(R) such that (&?lr/>
,(x - 2-‘n)),,Z
is an
orthonormal basis of O,-, The orthogonality of this family can be expressed with the Poisson formula
Let us write
(71)
(67)
We will show that this equation defines the Fourier
transform of a scaling function. We need to prove that
(a) 4(w) E L’(R) and (4(x - n)),,,Z is an orthonormal family.
(p) If V,, is the vector space generated by the family
of functions ( &, (X - 2:‘~)) nEZ, then the sequence of
vector spaces ( Vz I),i,z is a multiresolution approximation
ofL’(R).
Let us first prove property ((Y). With the Parseval theorem, we can show that this statement is equivalent to
Since &,(x)
orthonormal
E O2 I C V,, we can decompose it on the
basis ( 4 (x - n ) ) ,,Ez:
li/2~l(x) = ,,g,
(&I(U),
4(u
- n))
4(x - n).
(72)
Let us define
G(w)
=
E
( G2 i(u), $(u
,,= -cc
- n)) emino.
(73)
The Fourier transform of (73) yields
$(2w)
= G(w) J(w).
(74)
As in Appendix C, (74) and (71) give
Let us define the sequence of functions ( g,(w)),
that
g,(o)
=
for (w ( < 2qr
fi
H(2-“w)
P=l
for Iw[ 1 2qn.
0
, , such
Vn E Z,
5
erkwdw =
ar
i
ifk
ifk
(69)
Since H(w)
= H(w)
f a)(’ = 1.
(75)
each function of the family ( &?+2m1(x
- 2-‘n)),Ez.
The Poisson formula shows that this property is equivalent to
Vn E
Z,
5
q+(2w + 2n7r) $(2w
,1=--03
+ 2n7r) = 0.
(76)
= 0
# 0.
(70)
With hypothesis (17~) of Theorem 2, it is then possible
to apply the dominated convergence theorem to the seto derive from (70) that 4 (o) satisquence ( gq(w))y,l
fies (68).
Let us now prove property ( 0). In order to prove that
( W,,EZ is a multiresolution approximation of L’(R), we
must show that assertions (2)-(8) apply. The properties
(2)-(6) can be derived from the equation
$(2w)
+ (G(w
Since 02- 1 is orthogonal to VI I, each function of the family ( @$2m1(x
- 2-l n) ) ,rcZ should be orthogonal to
As q tends to + 00, the sequence (g,(u)),,
I converges
towards (4(w) (’ almost everywhere. We can also prove
+CZ
~Q1g,(w)
(G(wf
J(w).
satisfies (17a), we can show that
By inserting (64), (66), and (74) in (77), we obtain
H(w)
G(o)
+ r) G(w + T) = 0.
(77)
We can prove that the necessary conditions (75) and (76)
on G(w) are sufficient to ensure that ( fl&-~(x
2p’n)),,Ez is an orthogonal basis of 02-1. An example of
such function G(w) is given by
G(w)
= e -‘“H(w
+ 7r).
(78)
The functions G(w) and H(w) can be viewed as the
transfer functions of a pair of quadrature mirror filters. By
taking the inverse Fourier transform of (79), we prove
that the impulse responses ( g (n ) ) nEZ and (h (n ) ) ,lEZ of
these filters are related by
g(n)
From this equation, one can prove that the sequence of
vector spaces ( Vli)jGz defined in (0) do satisfy the last
two properties (7) and (8) of a multiresolution representation.
+ H(o
By definition,
satisfies
= (-I)‘-“h(
a multiresolution
lim Vz, = L’(R)
./++a
and
1 - n).
approximation
lim
p--m
(79)
of L’( R)
V,, = (0).
692
lEtE
TRAKSACTIONS
O N PATTERN
Since O? , is the orthogonal complement of V2, in V?, + I,
we can derive that that for any j # k, 02, is orthogonal
to O?h and
L’(R)
=
it
Oz,.
,,= -cc
(80)
We proved that for anyj E Z, (G&,(x
- 2-Jn)),,.z is
an orthonormal basis of 02, . The family of functions
( m~3r(X
- 2-/n)) C,I, IEZ: is therefore an orthonormal
basis of L’( R ).
E
APPENDIX
PROOF OF THEOREM
This appendix gives the main steps of Theorem 4 proof.
More details can be found in [35]. Let ( V2,)iGz be a multiresolution approximation of L’( R ) such that for any j E
Z,
(81)
approximation
where ( Vi, licz is a multiresolution
L’(R). We want to prove that the family of functions
(2-‘$&(x
2-‘&,(x
- 2p’ll) &,(y
- 2-h),
- 2-/n) &,(.v
- 2+n),
2-‘jc7iCer- 231) bi(Y
- 2-‘m))(,i,,,r)Ez2
8 vi,*,
This can be rewritten
v,,-1 = (V$, 0 Vi,) 8 (Vi, 0 Oil) 0 co;, 0 v;,)
The orthogonal
given by
0 o;,,.
complement
of V?, in VI,, I is therefore
02, = (Vi, 0 04,) 0 co:, 0 v;,,
0 (Oi, 0 o;,,.
(82)
The family of functions (m&,(x
- 2-.‘n))icz is an
orthonormal basis of Vi, and ( fl+*,
(X - 2-jn )),EZ is
an orthonormal basis of O:, . Hence, (82) implies that
(2-‘&,(x
- 2-‘n) I&,(J
- 2-/m),
2-‘&,(x
- 2-/n) &,(y
- 2:‘m),
2-‘$2~(x
-
-
2-‘n)
452iCY
INTELLIGENCE.
VOL.
II,
NO
7. JULY
1989
sum of the orthogonal spaces 02,
L’(R’)
=
ii
02,.
,=-LX
The family of functions
(2-&(x
2-‘4!3,(x
- 2-/n) il/z,(y - 2-Jm),
- 2-ln) &,(y
2-‘ic/n(,x - 231) h(Y
- 2-‘m),
- 2--‘4)(n ,,,)tZ’
basis of L2( R*).
ACKNOWLEDGMENT
I would like to thank particularly R. Bajcsy for her advice throughout this research, and Y. Meyer for his help
with some mathematical aspects of this paper. I am also
grateful to J.-L. Vila for his comments.
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= (Oi, 0 Vi,) 0 (04, 8 Vi,).
8 (o;,
MACHINE
of
is an orthonormal basis O?, . The vector space 02, is the
orthogonal complement of V?, in V2, + I. Let 04, be the
orthogonal complement of Vi, in Vl, I. Equation (81)
yields
V >,T’ = vi,+,
AND
is therefore an orthonormal
4
v,, = v;, 0 VA,
ANALYSIS
2-‘f4)(,i,,)i)Ezz
is an orthonormal basis of 02,
The vector sDace L’C R’) can be decomnosed as a direct
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Stephane G. Mallat was born jn Paris, France.
He graduated from Ecole Polytechnique, Paris, in
1984 and from Ecole Nationale Superieure des
Telecommunications, Paris, in 1985. He received
the Ph.D. degree in electrical engineering from
the University of Pennsylvania, Philadelphia.
PA, m 1988.
Since September 1988. he has been an Asaistant Professor in the Department of Computer Science of the Courant Institute of Mathematical
Sciences, New York University. New York. NY.
His research interests include computer vision, signal processing, and apnlied mathematics.
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