Discrete and Continuous Wavelet Expansions
YURI DEM’YANOVICH
Parallel Algorithm Department
Saint-Petersburg State University
Saint Petersburg
RUSSIA
LE THI NHU BICH
College of Education
Hue University
VIETNAM
Abstract: - This paper proposes a new approach to the construction of wavelet decomposition, which is suitable
for processing a wide range of information flows. The proposed approach is based on abstract functions with
values in linear topological spaces. It is defined by embedded spaces and their projections. The proposed
approach allows for adaptive ways of decomposition for the initial flow depending on the speed changes of
the last one. The initial information flows can be real number flows, flows of complex and p-adic numbers, as
well as flows of (finite or infinite) vectors, matrices, etc. The result is illustrated with examples of spline-
wavelet decompositions of discrete flows, and also with the example of the decomposition of a continuous
flow.
Key-Words: - wavelets, flows, decomposition, reconstruction, calibration relations
Received: April 16, 2021. Revised: January 3, 2022. Accepted: January 22, 2022. Published: February 23, 2022.
1 Introduction
The constant growth in the volume of numerical
information flows stimulates further development
and improvement employing their processing. The
structure and nature of these flows depend
substantially on areas of human activity (economics,
medicine, technology, etc.), the means and methods
of obtaining information, from the quality of
communication channels and their processing
speed.
Wavelet decomposition is one of the main
means of the processing of numerical information
flows. Its advantage over other processing methods
consists of the fact that the original numerical flow
is split into the main part, specifying (wavelet) part
and non-essential part. This explains the
widespread use of wavelet expansions in various
fields of human activity. Let us give several
examples of the application of these expansions in
technology and medicine.
In research [1] the separate models for signal
denoising with different ratio signal/noise were
designed. The discrete wavelet decompositions were
used. The result was applied to the computerized
analysis of Lung Sound.
Paper [2] is devoted to the damage severity
quantification of the brain by using a wavelet
packet. The proposed technique shows significant
benefits in compressing spatio-spectral patterns of
multichannel signals in just a unified visual
representation.
The timely and high-quality maintenance of
electrical networks is a prerequisite for their trouble-
free operation. In work [3], complex wavelets are
used to create an efficient algorithm for such
processing. The proposed algorithm achieves higher
accuracy with reduced training time in the
classification of events than compared to the
reported event classification methods.
As we can see from the above examples,
currently, the main problem is having insufficient
speed processing numerical information reaching
huge volumes. To solve this problem, a significant
advancement in the development of fast wavelet
algorithms decomposition is needed.
To date, there are several studies on the theory
of wavelets, among which special mentions are
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chilled works of I. Daubeshies [4], K. Chui [5], S.
Mallat [6], Yu. N. Subbotin and Chernykh [7], I.
Ya. Novikov, V. Yu. Protasov and M. A. Skopina
[8]. Research in this direction also includes a series
of modern works.
Cubic wavelets with two zero moments are
obtained in work [9]. Five-diagonal splitting for
cubic splines with six zero moments on the segment
was obtained in [10]. Paper [11] deals with the
structural issues that concern wavelet frames and
their dual frames. In paper [12] the authors define
the wavelet multiplier and Landau-Pollak-Slepian
operators on the Hilbert space. In paper [13] the
wavelet optimized finite difference B-spline
polynomial chaos method is proposed. The method
is applied to the solution of stochastic partial
differential equations. In paper [14] the authors
propose a highly efficient and accurate valuation
method for exotic-style options based on the novel
Shannon wavelet inverse Fourier technique
(SWIFT).
A new structure based on wavelet neural
networks, deep architecture and the Extreme
Learning Machine is proposed in paper [15]. The
proposed method is based on the Extreme Learning
Machine Auto-Encoder with a deep learning
structure and a composite wavelet activation
function used in the hidden nodes. Paper [16]
presents the rates of uniform strong consistency of
wavelet estimation for the nonparametric function in
the sup-norm loss by introducing an empirical
process approach. Paper [17] presents a new
construction of the homogeneous Dirichlet wavelet
basis on the unit interval. In work [18], a suitable
mother wavelet is selected for effective crack
detection in the beam. Paper [19] proposes a
parameter identification method of fractional-
order time delay system based on the Legendre
wavelet.
These studies mainly reflect the classic
approach to wavelets, which is based on various
variants of the Fourier transform, applied to the
multiple-scale ratio to obtain a scaling function and
ultimately wavelet decomposition.
However, the practice of processing numeric
flows required expanding the framework of the
classical theory. W.Sveldens constructed a lifting
scheme for an area that is not invariant relative to
the shift. The concept of non-stationary wavelets,
introduced by I.Ya. Novikov, also led to the
expansion the framework of the mentioned theory.
The need to significantly speed up computations
was faced with great theoretical difficulties that
arose on the path of the development of the classical
approach to the wavelet expansions.
The purpose of this paper is to consider another
spline-wavelet scheme decomposition. The
proposed approach is based on abstract functions
with values in linear topological spaces. It is defined
by embedded spaces and their projections. The
proposed approach allows for adaptive ways of
decomposition for the initial flow depending on the
speed changes of the last one. As initial
information flows can be real number flows, flows
of complex and p-adic numbers, as well as flows of
(finite or infinite) vectors, matrices, etc. The result
is illustrated with examples of spline-wavelet
decompositions of discrete flows, and also on the
example of the decomposition of a continuous flow.
The results obtained along this path lead to the
simple formulas decomposition and reconstruction.
In view of this, there is a significant acceleration of
computations. In the reviewed practical examples in
the case of rapidly changing flows, the computation
time is greatly reduced.
2 Auxiliary Assertion and Results
Let and
be topological vector spaces. We
denote conjugate spaces of linear continuous
functionals by and
accordingly. Let
be the value, which is the result of the action of the
functional on the element .
Analogously by (
we denote the result
of the application of the functional
to
element
.
Let be an interval In what follows, we
consider abstract functions , ,
with values in some vector space . If the closed
linear shell of values for the function
coincides with the space
,
then the function is
called a complete function.
Lemma 1. If is a complete function,
then for any nonzero functional from the
conjugate space there is a point
Proof. Proof by contradiction. Let be an
arbitrary nonzero element belonging . We
suppose that . By completeness
of we have {
}. Taking into
account the continuity of the functional , we
deduce that =0.This contradiction concludes the
proof.
Let and be complete functions of the
parameter with values in vector spaces and
respectively.
Consider vector spaces
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,
= { |
The fact that these spaces are vector follows from
easily verifiable properties: from additivity and
homogeneity with respect to operations of addition
and multiplication by a value.
Lemma 2. The following statements are true.
1. For function there is only one element
, for which representation is
true.
2. For function
there is only one element
, for which representation
is true.
Proof. Since both statements of the lemma are
stated under the same conditions with respect to the
spaces and
, then it suffices to prove only the
first of them.
We will prove by contradiction. Let there be
elements , , , for which
We put . According to (1), we find
point such that
Relation (3) contradicts formula (2). The received
contradiction proves the first assertion of the lemma.
The second statement turns out to be similar.
This concludes the proof.
Theorem 1. Let be a linear operator
:
If the formula
is right, then the space
is embedded in ,
Proof. By (4) we have
:
For each element
we find
so that
.Taking into account formula (5),
we have
By (8) we reduce
where . Thus, is an element of space
. Formula (6) has been established. This concludes
the proof.
Let the space
be embedded in the space
,
that is, relation (6) holds. Consider a projecting
operator , which projects the space onto
,
.This means that for the element there
are only elements and
such that
,
,
The element
is defined uniquely by .
Corresponding mapping is denoted by ,
It is easy to see that is a linear operator acting
from the space into the space
,
:
Thus, operator is defined by projector of
the space onto space
according to the
formula
(
Theorem 2. For any element ,
, the relation
is true.
Proof. From (7) and (9) we obtain
Passing to the adjoint operator in (14), we
obtain (13). This concludes the proof.
Let us introduce the operator . As
a result of projection (9), we obtain the direct sum
where
, .
Let . We put
Theorem 3. The relations
are right. Here .
Proof. From (13) - (14) and (16) we have
.
Thus, relation (17) is true. From (10) and (16) we
obtain (18). This completes the proof.
Item is initial flow, item is main flow and
is wavelet flow. Formulas (10), (16) represent
decomposition formulas, and formula (18)
represents reconstruction formula.
Theorem 4. The ratio
(19)
is right. Here is the identity operator in space
Proof. Since operator is the projector onto
space
, then it acts as the identical operator on
elements of this space. Thus, we have
On the other hand, the rule for calculation of the
projection has the form (see (14))
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Setting , from (21) we find
An obvious transformation of the left-hand side of
the last relation gives the formula
Using property (5) on the left-hand side of formula
(22), we obtain the equality of the left-hand side's
relations (20) and (22). Therefore, the following
formula is valid
In view of the completeness of the abstract
function from formula (23), we derive relation
(19). This concludes the proof.
3 On Embedding Spaces
Let be finite-dimensional spaces
of vectors with the scalar
product
.
Consider one more finite-dimensional space
of vectors
with scalar product
.
We consider the vector functions with a real
argument. We suppose that the closure of the linear
envelope of the set values of the functions
coincides with the entire space, i.e. in this section
we discuss complete functions.
Consider the complete functions and ,
, with values in the spaces
and , respectively.
We introduce linear spaces of functions by the
relations
Theorem 5. If the number matrix of size
has the property
,
then the space (25) is embedded in the space (24),
Theorem 5 is a special case of Theorem 1.
Example 1. Let , so the matrix
has the size . Suppose that
,
. We define the spaces
,
. It is clear that the relation
is right. By Theorem 1 the
relation
is fulfilled.
Example 2. Let , so the matrix
has the size . Suppose that
We define
and by formulas
,
. Let us
discuss the spaces
,
.
Obviously, the relation
is
right. By Theorem 1 the relation
is fulfilled.
4 Zero-type Wavelets in Finite-
Dimensional Spaces
4.1 Zero-order Splines
Let be an interval of . Consider a grid
.
Let be the set .We introduce
piecewise constant functions , [,
defined by the equalities
for ,
for ,
System of the functions (26) is
determined by the grid .
The functions are coordinate splines of
order zero.
For the space we take space of real vectors
with standard scalar product
.
In what follows, formula (27) is written in the
form
. Consider a linear space
defined by the relation
This space is called {\it generalized zero spline
space on the grid , and the elements of this space
are called splines of order zero.
Specifying the grid uniquely determines the
space .
Note that for fixed the sequence
contains one element equal to the unit. The rest of
its elements are equal to zero. So this sequence is
an element of space . So, is an abstract
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function with values in the space . That is why
the equivalence is valid
.
In view of this, formula (28) can be written in the
form . (29)
4.2 Embedded Space
Let be a natural number, . We discuss the
same half-open interval . Consider a grid
,
which is embedded in
,
.
Thus, for each we have
. Therefore, for
each there is a single number such that
. We denote this mapping ,
Let
. It is obvious that
. The
unambiguous inverse mapping is defined on
,
Similarly to the previous one, we introduce
piecewise constants functions , [,
defined by the equalities
for
,
for
,
System of the functions (32)
is determined by the grid
. The system is a
linearly independent system.
For the space
we take the space of
sequences =(
with the standard scalar
product,
.
Consider a linear space
This is the zero-spline space on the grid
.
4.3 Projection. Wavelet Decomposition
Let be the matrix of size
with elements
for [)
, for
[)
In the case under consideration, the relations
are true.
In the space , consider the system of linear
functionals defined by the ratios
It is easy to see that the system is
biorthogonal to the system of ,
Consider the projective operation
given by the formula
Considering that from
(35) we have
Introduce the matrix =( of size
with elements ,
. From relations (26) -- (29), (33) we
have
for ,
for , so that
Ratio (36) can be represented as
,
where
,
, .
Let us introduce the operator .
As a result of the projection, we obtain the direct
sum (15).
In the case under consideration, the
implementation of the decomposition formula is
obtained by substituting relations (36) and (37) into
representations (10) and (16). Using formulas (36)
in representation (18), we obtain the
implementation of the reconstruction formulas.
5 Illustration of Decomposition and
Reconstruction
As an illustration of the previous section’s results,
we consider the case when , . Let us
discuss the grid
In this case, we have . We
should consider six functions of the form (26)
for . For the space we take the Euclidean
space with the standard scalar product, namely
for vectors
with the standard scalar product
.
Defining the functions using formula (26),
Consider the space
As
consider the grid
,
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,where
,
,
.
We have , ,
,
Determining by formula (29), consider a
linear space
It is easy to check that
,
,
Thus =0 for
, for
for , .
According to the formula (36) we have
, for ,
, for ,
, for
In the case under consideration, the
decomposition formulas (15) - (16) take the form
, , , ,
,
, ,
Since ,
,
,
,
,
and
, then the reconstruction formulas (18)
can be written as
6 Zero-type Wavelets in Infinite-
Dimensional Spaces
6.1 Zero-splines
On the interval of the real axis we order
consider the grid
with properties
(39)
We introduce piecewise constant functions ,
t(), defined by equalities
for ,
for
System functions (40) are
defined by the grid (38) -- (39). This system is a
linearly independent system.
For the space , we take the space of real
sequences
with standard scalar product
.
Consider a linear space
(42)
Note that for fixed t() the sequence
contains one element equal to the unit. The rest of
its elements are equal to zero. So this sequence is an
element of the Hilbert space . We can discuss
as an abstract function with values in the Hilbert
space . That is why the next equivalence is valid
In view of this, formula (42) can be written in the
form
6.2 Embedded Space
On the same interval (), consider the grid
with properties
Thus, for each we have
Therefore, for
each there is a unique such that
.
We denote this mapping ,
Let be the set . It is obvious that
the single-valued inverse map is defined on
Similarly to the previous one, we introduce
piecewise constants functions
, , defined by the ratios
for
,
for
,
For the space
we take the space of real
sequences
with standard scalar product
.
Consider a linear space
Thus we have the zero-spline space on the grid
.
Let be the matrix of infinite size
with elements
for [)
, for
[)
In the case under consideration, the relations
are true.
6.3 Projection onto Embedded Space
In the space , consider the system of linear
functionals defined by the ratios
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It is easy to see that the system
biorthogonal to the system of ,
Consider the projective operation
given by the formula
Note that there are no questions about the
convergence of the sum in (47), because for every
fixed t() in this sum there is only one nonzero
term. Considering that from
(48) we have
We introduce the matrix =( of infinite
size with elements , . From
relations (40), (43) -- (46) we have
for , for , so that
Ratio (48) can be represented as
where
,
,
. (50)
6.4 Wavelet Decomposition
Let us introduce the operator . As a
result of projection (47), we obtain the direct sum
S=
where
We introduce the notation
By Theorem 3, we obtain ratios
,
Formulas (50) - (52) illustrate the wavelet
decomposition.
Thus, the initial flow spawns two flows:
main flow and wavelet flow . Relations (50) -
(51) are the decomposition formulas. Ratio (52) is
the reconstruction formula.
7 About Continual Wavelets
7.1 Generating Function
To illustrate the possibilities of the proposed
approach, consider a continuous wavelet
decomposition.
Let be the Lebesgue measurable set of the real
axis . We discuss point
t
of an interval .
In the case of measurable sets of the
symbol , we agree on use with the
meaning "for almost all ". For example, the
expression means that the
function
is equal to zero at all points of the
set , excluding some set of measure zero.
Consider the function
We suppose that for each fixed the element
belongs to the space. Here
is fixed
number, . In this way,
is a trajectory in the space. In
what follows we assume that the functions
depending on are defined for all points of the
interval .
According to the general scheme (see Section 2.) we
put , ,where
.
Consider linear space S defined by the relation
S={u(t) | u(t)=
Remark 1. The zero-order wavelets considered in
the previous section are a special case of the
situation under consideration here. In particular,
using grid (38) - (39), we can define the function
in the following way. For each
and we put
for , for
S={|=(c,
7.2 Embedded Space
Consider a measurable set contained in
Consider the function
t which for each fixed t is an
element of the space . Thus ,
t is a trajectory in the space . In
accordance with the general scheme, we put
Let the linear space
be defined by
Consider the function
,
with properties
We suppose that
It follows from (53) that the operator :
defined by the formula
[v()]
has the property
In accordance with Theorem 1, relation (54)
implies embedding of the space
into the space
,
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In what follows, we consider the projection of the
space onto
7.3 Embedded Space Projection
As an illustration, we consider a simple example of
a projection operation from the space to the space
.
In what follows, we use the concept of the
restriction of a function from the set to the set .
The restriction of the function from to
is called function such that
, and .
The restriction of from to is
denoted by the symbol . In this way, we have
We propose that functions are
propagated from onto by zero. Therefore,
we can discuss the embedding
Let be the operation of the restriction from the
set onto the set
[
()]()=
(
In the case under consideration, the operation
the restriction to . Therefore, formula (57) takes
the form
By (58) we see that the values of on the
set does not play a role. So in view of the
arbitrariness of = the integral on the
right parts of the above equality can be calculated
only over the set . We have
Identity (59) implies the formula
for
for
We suppose that
By (61) formula (55) follows from Theorem 1
and relation (56).
To define the operation , we consider an
arbitrary element . By definition the space
has the form
, where
the function lies in the space .
Obviously narrowing of this function to
the set lies in the space
By the image of the function under the
mapping we consider the function
, where
In the previously adopted notation, this can be
written with the ratio
=(
where .
A shorter record of the last relation is
=
=(
Obviously, the operator , defined by formula (62),
is linear operator. The last one acts from to
Let us show that the operator is idempotent, that
is, the relation
is true. It suffices to
establish that on the space
the operator acts as the
identity operator. Indeed, according to the
definition of the operator , we have
=
=
Instead of , we substitute the function
. Now the integral on the left-hand
side of formula (63) can be calculated only over
the domain , so we have
=
=
Taking into account relation (61), we arrive at
the identity
Thus, the operator
is the projector.
From formulas (12) and (63) in the case under
consideration, we obtain
=
=
Using the completeness of the abstract function
, we arrive at equality
7.4 Wavelet Decomposition
Let's introduce the operator . As a
result of projection (64), we obtain the direct sum
, where
, . Let us turn
to the decomposition formulas (10), (16) and (65) in
the case under consideration.
The main flow is obtained by the formula
So in this case of main flow is determined
by the values of the original flow on the set
,
The wavelet flow is obtained according to the
relations
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Using formula (60), we find
for
for . (67)
Thus we obtain decomposition formulas (66) -
(67).
For the formula (18), we take into account
relation (60), (67).
We have
for
for
Therefore we find
for
for (68)
Ratios (68) are reconstruction formulas in the
discussed case.
8 Conclusion
In this paper, a new approach to the construction of
wavelet expansions is considered. This approach
allows us to consider discrete and continuous flows
of a various nature. Thanks to this approach, it is
possible to consider discrete numerical flows, flows
of matrices, flows of p-adic numbers, etc.
The possibility of wavelet decomposition of
continuous streams seems to be very important,
which makes it possible to include in the processing
not only discrete (in particular, digital) signals, but
also streams representing analog signals. The result
obtained here can be viewed, on the one hand, as the
limiting closure of the algorithm (in the sense of
S.L. Sobolev), and on the other hand, as a source of
discrete wavelet approximations (see Remark 1.)
The need for a very brief presentation of the
material forced the authors to limit themselves to the
simplest examples illustrating the proposed
approach. However, the examples provided suggest
broad application prospects. The authors intend to
investigate some of these applications in the future.
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.9
Yuri Demyanovich, Le Thi Nhu Bich
E-ISSN: 2224-2880
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Volume 21, 2022
