1. Introduction
Wavelet systems are widely used in applied mathematics for signal analysis and synthesis, compression tasks, data preprocessing and feature extraction for neural networks and so on in a large number of applications (see, for example, [
1,
2,
3,
4,
5,
6]). The symmetry property is among the most desirable features for wavelets. Multiwavelets are a natural generalization of usual wavelets (see, for example, [
7,
8,
9] for the pioneering works and [
10] for the comprehensive treatment of the subject). Their construction is more flexible since multiwavelets can provide similar to usual wavelets properties but with shorter support or even provide incompatible for usual wavelets properties. For instance, in [
11], examples of univariate refinable function vectors with dilation factor
, which are interpolating, orthonormal, continuous and with compact support simultaneously, were constructed. The paper is devoted to the construction of symmetric multiwavelet systems in the multivariate case with several other useful properties such that desirable number of vanishing moments and high order of the balancing property.
Let us briefly highlight current investigations on the problem of the construction of symmetric multiwavelets. In the univariate case, general methods for the construction of orthogonal and biorthogonal multiwavelets with symmetry were suggested in [
12,
13]. In the multivariate case, the development of the analogous general methods is a much more difficult task, due to the complexity of the matrix extension problem (see
Section 4 for details). General results related to the multivariate extension principle for multiwavelets can be found, for example, in [
14,
15,
16] and the references therein. The symmetry properties of wavelets for the multivariate case can be defined using the notion of symmetry groups (for details see
Section 2 and also, for example, [
17]). For a simple symmetry group
symmetric multiwavelet systems based on lifting scheme were constructed in [
18]. Additionally, for the same symmetry group, the construction of symmetric multiwavelet frames as a symmetrization of some multiwavelet frames was suggested in [
19]. For a class of appropriate (in some natural sense) symmetry groups
, a numerical method for the construction of interpolating refinable function vectors and their duals with the
-symmetry property, based on solving certain linear systems of equations, was given in [
20]. Additionally, a method for getting multiwavelets was suggested in [
20], but there were no proofs concerning the properties of the obtained multiwavelet systems, such as symmetry and vanishing moments. A family of symmetric dual multiwavelet frames obtained by convolving scalar wavelets with multiwavelets from [
20] in a Kronecker type manner was produced in [
21]. To the best of the author’s knowledge, no other approaches related to the construction of symmetric multiwavelets in the multivariate case were suggested.
The results of the paper propose a method for the construction of dual multiwavelet frames, starting from symmetric interpolating refinable function vector in the multivariate case. The construction starts from a pair of appropriate symmetry group
and matrix dilation
M (there are some restrictions on the choice of this pair in the interpolating case, see
Section 3 for details). The initial symmetric interpolating matrix mask can be constructed using several methods developed in [
20] or [
22] (see also [
23]). Any preassigned order of the sum rule can be provided for the matrix mask (this property is related to approximation properties of the resulting multiwavelet system). Next, a symmetric dual matrix mask is constructed in two steps. Firstly, a dual matrix mask is constructed using the algorithm in [
24]. Further, a symmetrization step is completed. Again, any preassigned order of sum rule can be provided for the dual matrix mask. Finally, the matrix extension principle is used for the construction of wavelet matrix masks. The main point here is to extend two known blocks of rows with trigonometric polynomials up to square matrices of trigonometric polynomials such that the columns of these matrices are biorthogonal. Generally speaking, multivariate matrix extension is not a simple process. However, in the interpolating case, matrix extension can be done explicitly and the symmetry properties for wavelet functions can be provided.
The paper is organized as follows. In
Section 2, basic notations and definitions are given, including the description of the action of a symmetry group on a set of digits, which is a key point in the description of symmetry properties for wavelet functions and their masks.
Section 3 is devoted to the facts related to the construction of
-symmetric refinable matrix masks with a desirable order of sum rule. This is the starting point in the construction of multiwavelets. In
Section 4.1, the construction of
-symmetric dual matrix masks with a desirable order of sum rule is presented.
Section 4.2 describes the construction of dual multiwavelet systems with symmetry properties. In
Section 4.3, the order of vanishing moments and the balancing property of the obtained multiwavelets are discussed together with some other aspects. In
Section 5, several examples are presented.
2. Basic Notations and Definitions
For , their inner product is denoted by , means that for , , If , , we set , , , , is the Kronecker delta. For , For , The cardinality of a finite set is denoted by . The (i,j)-th element of matrix M is denoted by The i-th column of matrix M is denoted by . denotes the set of sequences on with finitely many non-zero elements.
An integer
matrix
M is called a dilation matrix if the eigenvalues of
M are greater than 1 in absolute value.
,
denotes the
identity matrix,
A complete set of representatives of
is denoted by
. An element of
is called a
digit,
is called
the set of digits (see, for example, [
25] Section 2.2). We assume that
The coset corresponding to digit
is denoted by
, i.e.,
By
, we denote the set of cosets. The set of digits will be frequently used as the index set for some objects, so we assume that the digits are ordered somehow such that the first digit is
.
The Fourier transform of a function is This notion can be naturally extended to and to tempered distributions. For a column vector of functions , we set , if applicable.
Additionally, the following lemma, which can be proved by direct computations, will be useful.
Lemma 1. Let A be a invertible real-valued matrix, For any sufficiently smooth functions f and g defined on , the following statements are equivalent
- 1.
for all ,
- 2.
for all ,
- 3.
for all .
Action of a Symmetry Group on a Set of Digits
A finite set of unimodular matrices (i.e., integer matrices with determinant equal to ) is a symmetry group with respect to the dilation matrix M, if forms a group under the matrix multiplication and for all In what follows, we assume that is a symmetry group with respect to some fixed dilation matrix M.
A function
is called
-symmetric with respect to the center
, if
For trigonometric polynomials, we use a slightly different definition, which is compatible with the above definition of symmetric functions, if we are dealing with compactly supported refinable functions.
A trigonometric polynomial
is
-symmetric with respect to the center
, if
where
for all
Condition (
1) is equivalent to the fact that
and
We say that
is an
appropriate symmetry center for
, if
for all
.
It can be checked that for any
, for any coset
and any appropriate symmetry center
for
, there exists a unique
such that
. This fact allows us to define a group action on the set
by the group
(associated with the symmetry center
c) as follows (for details see, for example, [
26,
27])
where
is such that
Further, we need some notions and results from group theory (see, for example, [
28]). The
orbit of
is
. Clearly,
and two orbits are either equal or disjoint. Denote by
the set of representatives from each orbit, then
For convenience, redenote the elements of the set
by
where
such that
For a fixed index the stabilizer of is denoted by Note that and is a subgroup of A left coset of by subgroup is a set where . Any two left cosets are either coincide or disjoint. denotes the set of left cosets of by .
Let
be a set of representatives of
Then
. Thus, the orbit
can be represented as
For a fixed index
p, matrices of the set
will be denoted by
,
We assume that
Define
Then
and the elements of the orbit
are
where
This redefined after the division by orbits set of digits is called to be associated with the symmetry center
c and is denoted by
.
For fixed , by the Lagrange’s theorem and for each matrix K in , there exist unique matrices and such that The sets can be considered the "coordinate axes" of the symmetry group (for each these “coordinate axes” of can be different).
Fix index
. For a matrix
consider
. There exist matrices
and
such that
. Then
. Next, since
there exist matrices
and
such that
. Then
. Here the notation
means the map from the set of indices
to itself and the index
is uniquely determined by index
i and matrix
for a fixed index
Additionally, by
, we denote a special vector from
such that
For convenience, a brief summary of notations related to the action of symmetry group on the set of cosets
is presented in
Table 1.
3. Symmetric Refinable Function Vectors
Consider compactly supported functions
,
and denote
.
is called a
refinable function vector if it satisfies a refinement equation
where a finitely supported sequence of matrices
is called a mask with
multiplicityr. The
symbol of sequence
is defined by
i.e.,
is a matrix of trigonometric polynomials, which we will call a
matrix mask. If the Fourier transform is applicable, the refinement Equation (
2) can be equivalently written as
It is known ([
29] Proposition 1) that the compactly supported distributional solution of the refinement equation is unique (up to multiplication by a constant), when matrix mask
consists of trigonometric polynomials and matrix
has only one eigenvalue equal to 1 and others are less than 1 in absolute value.
A refinable function vector
is
-symmetric with respect to the row of centers
, if each function
is
-symmetric with respect to the center
. To define the
-symmetry property for matrix masks, we need an additional notation. Suppose
and denote
The obvious properties of
are
,
,
. Then the
-symmetry property of
in terms of the Fourier transform can be written as
Together with the refinable Equation (
3) this yields that
On the other hand, since
, we can write
This is the motivation for the definition of symmetric matrix masks. We say that matrix mask
is
-symmetric with respect to the row of centers
, if
Or, equivalently, the
th element of the matrix mask
is
-symmetric with respect to the center
, since
Of course, it should be provided that
. But we require a little more. Namely, we say that the row of centers
is
appropriate for , if
for all
,
A connection between the
-symmetry of refinable function vector and its matrix mask is stated in the following theorem (see [
20,
22]).
Theorem 1. Let be an appropriate row of centers for . Suppose is the unique compactly supported solution of the refinement equation with matrix mask and the multi-integer shifts of refinable function vector Φ are linearly independent. Then Φ is -symmetric with respect to the row of centers C if and only if matrix mask is -symmetric with respect to the row of centers
Here, the linear independence of the multi-integer shifts of means that the synthesis operator is injective, where is defined on as follows for .
Let
be a matrix mask. We say that
obeys
sum rule of order n (with respect to the dilation matrix
M) if there exist a
row of functions
such that
,
and
In fact, to formulate and check these conditions, we only need to know the derivatives of
y at the origin up to the order
n. However, in the paper, it is more convenient to write and apply these conditions using the row of functions
y.
A matrix mask
with sum rule of order
n implies the accuracy of order
n for the corresponding refinable function vector
, which means that polynomials up to order
n lies in the shift-invariant space generated by functions
. In more details, suppose that
is a refinable vector of compactly supported distributions related to matrix mask
and the sequences
, are linearly independent for all
Then the accuracy of order
n for
is equivalent to the sum rule conditions of order
n for
(see ([
24] Theorem 2.4) for details). Note that the above conditions about linear independence will be valid if, for example, the multi-integer shifts of
are linearly independent, since this fact is equivalent to for examplelinear independence of the sequences
,
, for all
(see, for example, ([
30] Theorem 5.1)). The order of accuracy of
is related to the approximation order of the corresponding shift-invariant space (see, for example, [
31]).
The
polyphase component of the mask
A corresponding to the digit
is a sequence
defined by
The
polyphase representation of the matrix mask
is
However, it is more convenient to use another partition for the polyphase components. Consider the
-th column of
and place the columns of its polyphase components into a matrix
for
In other words, the
-th column of
is the column with index
s in the matrix
, i.e.
for
We call these matrices
,
the modified polyphase components of
. Note that the following representation is valid
3.1. Symmetry of the Polyphase Components
Next, we need to reformulate the
-symmetry conditions of a matrix mask in terms of its modified polyphase components. Let us fix an appropriate row of symmetry centers
for the symmetry group
Suppose
is an
-symmetric matrix mask, i.e., (
4) is valid. Consider this condition by columns:
where
is the
-th column of matrix
,
. Note that the certain columns of matrix mask
turn out to be mutually symmetric and this can be described. Recall that matrix
contains the polyphase components of the column
. It is convenient to index the columns of modified polyphase component
using the digits, while digits are indexed according to their division by orbits with respect to the group action
where
is such that
The corresponding notation for set of digits associated with the symmetry center
is the following:
The “coordinate axes” for will be denoted by Additionally, later we have to deal with several columns of matrix mask at the same moment, so it will be convenient to use notation for digits from .
Lemma 2. A matrix mask is -symmetric with respect to the appropriate row of centers if, and only if, for any fixed and for each the columns , satisfywhere j is such that for matrices and , is such that Proof. Suppose
is
H-symmetric with respect to the appropriate vector of centers
. Let us fix
Then by (
7) and (
10) we obtain for the
-th column of
and
that
where index
j is connected with
i by
, i.e.,
j is such that
for matrices
and
. On the other hand, we have
Since the polyphase representation is unique with respect to the chosen set of digits
, we obtain (
9). The converse statement can be proved by the analogous straightforward computations. □
In particular, for a fixed
and
for any
we obtain
where
This leads to the fact that matrix element
is
symmetric with respect to the center
Additionally, by the choice of
, we obtain
for
,
By direct computations, it can be checked that matrix element
is
symmetric with respect to the center
Here
is a conjugate subgroup to
by matrix
, namely
where
,
3.2. Interpolating Refinable Function Vector
A refinable function vector
is called
interpolating, if multiplicity
r is equal to
, all components of
are continuous functions with compact support and
Since
, we can associate
with corresponding digit
and we can index functions
using digits. Condition (
13) can be rewritten as
Or, since any
can be represented as
for some
condition (
14) is equivalent to
An interpolating refinable function vectors related to Shannon-like sampling expansions
This expansion allows us to state that the multi-integer translates of
are linearly independent. Indeed, if
is the synthesis operator of
, then
for
implies that
by (
16).
For an interpolating refinable function vector its matrix mask should have the following special form of the first column: for . The following theorem states the details.
Theorem 2 ([
32] Theorem 2.1)
. Let Φ be a refinable function vector with matrix mask . Then Φ is interpolating refinable function vector if and only if- 1.
;
- 2.
;
- 3.
andwhere are some trigonometric polynomials, .
The quantity
is related to
-smoothness exponent of a refinable function vector
with dilation matrix
M and matrix mask
A (see [
32] for details). Matrix mask
in (
17) is called an interpolating matrix mask.
Let us consider an
-symmetric interpolating refinable function vector
with the row of symmetry centers
From the
-symmetry property and (
15) we have
for all
and
Then
Hence for all and . This condition will be satisfied, if for all . So, in the interpolating case the symmetry centers are defined by the digits.
By Theorem 1 for the
-symmetric interpolating refinable function vector
with matrix mask
matrix elements
,
are
-symmetric with respect to the centers
or
Thus, the natural limitation on the symmetry centers for trigonometric polynomials gives the following restrictions
The latter condition reduces possible variants for the appropriate choice of symmetry groups
and dilation matrices
M, i.e., (
18) should be checked for a chosen pair
and
M before the construction.
Additionally, note that if matrix mask
of an interpolating refinable function obeys sum rule of order
n, then
y is not arbitrary. For instance, the appropriate choice is
(see, for example, [
33]).
Methods for the construction of
-symmetric matrix masks with any preassigned order of sum rule were suggested in [
20,
22]. Additionally, the canonical form of matrix mask (see, for example, [
34]) can be used for the construction of symmetric matrix masks starting from any appropriate refinable mask in the scalar case (see, e.g., [
22] for details). For instance, any known interpolating symmetric refinable masks can be used (see, for example, [
22,
27,
35,
36,
37] for examples of such masks and for methods of their construction). This provides the initial step in the construction of symmetric multiwavelets.
4. Extension Principle
A general scheme for the construction of compactly supported MRA-based wavelet systems was developed in [
38] (the Mixed Extension Principle). This scheme is also applicable for the construction of multiwavelet systems.
Let
,
be compactly supported refinable function vectors, which satisfy the refinement equation with matrix masks
. For convenience, we use notations
,
. Assume that there exist
matrices of trigonometric polynomials
,
,
, such that the following
matrices of
blocks
satisfy
Wavelet function vectors
,
, are defined via its Fourier transforms
Then the set of the function vectors , is said to be a compactly supported MRA-based dual multiwavelet system generated by the refinable function vectors (or their matrix masks ).
The set of function vectors
is a multiwavelet frame in
, if there exist positive constants
and
such that
Here,
and
. We say that a pair of sets
and
is a dual multiwavelet frame in
, if each set is a multiwavelet frame in
and
where the series converge unconditionally.
It is known (see, for example, ([
15] Theorem 2.1)) that compactly supported MRA-based dual multiwavelet system
,
is a dual multiwavelet frame in
if
,
and
,
,
(i.e., all wavelet function vectors have vanishing moments at least of order 1).
Since wavelet matrix masks , are matrices of trigonometric polynomials, then the wavelet function vectors , , are linear combinations of function vectors , respectively. Therefore, wavelet functions in vectors , also have compact support and they are in the same function space as functions from refinable function vectors.
For the realization of the extension principle starting from some initial symmetric matrix mask
, we need to construct symmetric dual matrix mask
and we need to obtain symmetric wavelet matrix masks
such that the equality (
20) is valid. In this paper, we set
, i.e., the extended matrices
and
are square block matrices.
4.1. Symmetric Dual Matrix Mask
Firstly, we need to obtain a symmetric dual matrix mask. Let
be a matrix mask. For a finitely supported sequence of matrices
its symbol
is called a dual matrix mask for
, if
A dual matrix mask for
with preassigned order of sum rule can be constructed using the following result (see ([
24] Theorem 3.4) for the constructive proof).
Theorem 3. Let be a matrix mask such that has at least one dual matrix mask. Then for any , there exists a dual matrix mask which obeys sum rule of order .
Note that for the interpolating matrix mask
, at least one trivial dual matrix mask always exists (see
Section 4.2). In order to provide the
-symmetry property for the dual matrix mask
we need several additional considerations. Let
be a dual matrix mask, constructed by Theorem 3. Its symmetrization can be done as follows:
Below, we prove that
, indeed, is a dual matrix mask for
with the
-symmetry property and with the same order of sum rule as
.
Theorem 4. Matrix mask is an -symmetric dual matrix mask for an -symmetric matrix mask .
Proof. First, let us show that
is
-symmetric with respect to the same row of centers
C as the primal mask
. For
, consider
Thus,
is
-symmetric with respect to the same row of centers
C. Next, we show that
is a dual matrix mask for
, i.e.,
Inserting the definition of
into the left-hand side of (
23), using (
4) for
and the following equality
the left-hand side of (
23) is equal to
It remains to note that the inner sum over
can be rewritten as
since
and
for some
,
. Thus, if
q is running over
, i.e., over the complete set of representatives of
, then
is also running over the complete set of representatives of
, but may be in another order and may be across some other representatives. Nevertheless, due to the periodicity of
we get the equality
and the same for
Therefore, since
is a dual matrix mask to
, then the right-hand side of the last formula is indeed equal to
. □
Next, we check the new dual matrix mask
has the same order of sum rule as the old dual matrix mask
. If the dual matrix mask obeys sum rule of order
, then Equations (
5) and (
6) for
are valid with some row of trigonometric polynomials
. However,
cannot be arbitrary. It is known that the derivatives
for
are uniquely defined (up to a multiplication by a constant) and can be found based on the equality (
22). This fact is stated in Theorem 3.1 in [
24]. In more details,
should satisfy the following conditions:
or equivalently
Theorem 5. Let be an -symmetric matrix mask and let be its dual matrix mask with sum rule of order . Then matrix mask obeys sum rule of order .
Proof. The fact that equality (
25) is valid with symmetric matrix mask
imposes some conditions on the choice of
. Indeed, from the one hand (
25) can be rewritten as
for
. On the other hand, by the
-symmetry of
, we obtain
Or by Lemma 1, the last equality is equivalent to
Comparing Equations (
26) and (
27) and using the fact that the derivatives of
are uniquely defined by (
25), we get that
Since only the derivatives of
are needed, we can assume that
Next, show that
also obeys sum rule of order
with the row of trigonometric polynomials
. Consider,
Next, for any
conditions
by Lemma 1 are equivalent to
Additionally, by (
24) and (
28) we have
Therefore, we obtain that
by the fact that
, where
is such that
and, thus, (
29) can be applied. □
4.2. Symmetric Wavelet Matrix Masks
Let
be a symmetry group with respect to the dilation matrix
M, such that
for all
and for all
. Let
,
, be an interpolating refinable function vector which is
-symmetric with respect to the centers
,
Let
be the corresponding matrix mask. Then mask
is also
-symmetric, i.e., (
4) is valid or the matrix element
is
-symmetric with respect to the center
,
Let
be an
-symmetric dual matrix mask.
It is more convenient to write the process of matrix extension using the modified polyphase components. First, note that for the modified polyphase representation of each column of matrix mask
, we will use its own set of digits. Namely, for a fixed
, consider the set of digits
associated with the symmetry center
. In what follows, the modified polyphase components of the column
of
are defined using the set of digits
. The modified polyphase representation of
defined in (
8) can be alternatively written as
where
is the matrix defined by
It is known that
is a unitary matrix, i.e.,
. Denote by
a block-diagonal matrix
. Clearly,
is also a unitary matrix. Let
be the standard basis in
. Then
where
is the Kronecker product of matrices
A and
B.
Now, return to the extended matrices
and
defined in (
19) with
. Since these matrices are square matrices, then
is equivalent to
. Consider the first block row of matrix
and denote it by
. Additionally, denote the row of matrices
, by
These matrices are connected by
matrix
, i.e.,
Lemma 3. Let A and be in . Define the rows , byand , , where and , are the modified polyphase components of and , respectively. Then or
Proof. Note that matrix
V is a unitary matrix since it is a product of unitary matrices
and
, where the latter matrix is just a rearrangement of columns of
. Hence
and
□
This lemma allows us to extend matrices
and
from the initial first block row in terms of the modified polyphase components. Consider the row of the modified polyphase components of matrix mask
. Due to Theorem 2 its first block has a special form:
. This consideration easily implies the fact that interpolating matrix mask
always has at least one dual matrix mask, since the row of the modified polyphase components of such dual matrix mask can be taken as
. Denote
. Next, let
be a dual matrix mask constructed in
Section 4.1 and consider its row of the modified polyphase components. Denote
. These two block rows of the modified polyphase components are biorthogonal by design. Starting from these rows, the required matrix extension up to square matrices can be done explicitly as follows
By direct computations, it can be checked that
. These matrices contain the modified polyphase components of all masks, and they can be transformed back into matrices
using the transformation matrix
V. By Lemma 3,
is equivalent to
. The next theorem describes the symmetry properties of the obtained by (
30) wavelet matrix masks.
Theorem 6. Let be an interpolating -symmetric matrix mask with respect to row of centers , where . Let be a dual -symmetric matrix mask with respect to the row of centers C. Let , , be wavelet matrix masks, obtained via the matrix extension based on (30). Then wavelet function vectors and contain wavelet functions, which are mutually symmetric in the following sense. For fixed each wavelet function from the vector can be associated with a digit from , i.e., can be associated with , , same for . Then for where j is such that for the group action associated with the symmetry center and is defined in (10). Proof. Let
denote the modified polyphase component of
, namely,
contains the polyphase components of the
-th column of matrix mask
. Additionally, let
and
. Using these notations, the matrices in (
30) can be written as
Next, we check the symmetry properties of wavelet matrix masks and then show how these properties will reflect on wavelet function vectors.
Let us start with matrix
. Fix
and the corresponding set of digits
. Consider block
in matrix
and its elements
here
. By (
30)
. According to Lemma 2 for the column with index
of matrix
, we have the following relation for
:
where
j is such that
for matrices
and
,
is defined in (
10). For the column elements, after applying the complex conjugation, we obtain
Since
, then
. Hence, changing
by
, we can rewrite
Summing over
and using the modified polyphase representation, we can get the first column of the wavelet matrix mask
. Namely, since
then (
32) implies that
where
j is such that
with the group action associated with the symmetry center
and (
10) is used.
The other columns of wavelet matrix mask
, are equal to
except the
-th column, since the corresponding block with the modified polyphase component
is equal to
Then
-th column of wavelet matrix mask
contains elements
,
. It’s not hard to see that
Overall, we obtain the following relation:
which is true for
by (
33), since
, and for
by (
34), and also for all other
since the elements in the
l-th column,
, are just zeros.
Next, let us obtain the analogous relations for wavelet matrix masks obtained from the matrix
. The blocks
obviously gives the result similar to (
33), since the symmetry properties of the elements of the rows
P and
are the same. Therefore,
where
j is such that
with the group action associated with the symmetry center
.
Next, consider other blocks of matrix
. Note that each block has the form
,
. Consider each block element-wise, where elements are indexed by double index as follows: the rows are indexed using
, i.e., using indexing of the digits in
and the columns are indexed using
, i.e., using indexing of the digits in
. Using the symmetry properties of the modified polyphase components
and
from Lemma 2, we obtain
where
is such that is such that
with the group action associated with the symmetry center
and
such that
with the group action associated with the symmetry center
,
is defined as
and similar for
.
Next, consider the modified polyphase representation in order to obtain the elements from the
-th column of wavelet matrix mask
where the sum is taken over all indices of digits
from
. Then relation (
36) implies that
or
This gives the required symmetry relation similar to (
35).
Next, we check how the obtained symmetry relation for wavelet matrix masks will affect on wavelet function vectors. Recall that
Since only two columns of
are non-zero, then
By default,
However, it is more convenient, if wavelet functions are indexed in the same way as digits from
. Namely, consider
, where
. Then for
equality (
35) implies that
Or
where
j is such that
with the group action associated with the symmetry center
. This means that functions of wavelet function vector
are mutually symmetric with each other.
Similar relations can be established for wavelet functions in vector
. Fix
, where
. Then for
equality (
37) implies that
where
j is such that
with the group action associated with the symmetry center
. □
Relations (
31) describe the symmetry properties of the resulting wavelets. If the set of digits
associated with symmetry center
is such that all orbits contain only one element, i.e.,
, then all wavelet functions are, indeed,
-symmetric. If some orbit has more than one element, then the corresponding wavelet functions are mutually symmetric, which means that for some matrices
, one wavelet function transforms into another according to relations (
31). For some symmetry groups, additional symmetrization step can be done, which helps to obtain truly
-symmetric wavelets from mutually symmetric ones. The idea of this symmetrization step and details can be found in [
27].
4.3. Further Details and Discussion
The above considerations give a method for the construction of compactly supported MRA-based dual multiwavelet system with symmetry properties. It remains to check that the obtained system of multiwavelets is, indeed, a dual multiwavelet frame. Recall that functions in vectors
and
should be in
and vanishing moments at least of order 1 for all wavelet functions should be provided. The fact that refinable functions belong to
can be checked after the construction of matrix masks. Moreover, it is known (see ([
39] Theorem 2.2)) that if compactly supported refinable functions are in
, then they are in
with some
. This value
can be computed using the algorithm in [
34]. Regarding the vanishing moments of wavelet functions, the following lemma, which is the generalization to the multivariate case of Theorem 2.1 in [
40], will be helpful.
Lemma 4. Let and be the matrices defined in (19) with . Then obeys sum rule of order n with a row of functions y if and only if Proof. The sum rule condition of order
n for matrix mask
can be rewritten as follows
Since
then
Multiplying the last equality from the left by the column vector
, we obtain
Taking the derivatives of order
at the origin, we obtain
By (
38) and since
we obtain
Conversely, since
is equivalent to
, then
Multiplying the last equality from the left by
, we obtain
Taking the derivatives of order
at the origin, we obtain
which is equivalent to sum rule condition of order
n. □
This result allows us to check that MRA-based multiwavelets are provided with vanishing moments of order 1. Indeed, if the initial matrix mask
obeys sum rule at least of order 1, then Lemma 4 says that
Note that and is the unique 1-eigenvector of , therefore . Hence, is equal to up to a multiplication by a constant and By duality, it is also true that , if the dual matrix mask obeys sum rule at least of order 1.
Moreover, in practice, dual wavelet function vectors
are used for the analysis operations, and it is good to provide these functions with high order of vanishing moments. This is indeed the case, namely, the order of vanishing moments for
is equal to
n (i.e., equal to the order of sum rule of the initial matrix mask
). Note that by Lemma 4
We have to check that
, since this equality provides vanishing moments for
. Indeed, equalities
imply that
,
By the refinement equation, it is true that
Note that by duality and by Equation (
25), the derivatives of
y at the origin are related with the derivatives of
as follows
Since this relation defines the derivatives of
y at the origin uniquely (up to a multiplication by a constant), then comparing it with (
39), we obtain the required connection
.
It is also worth noting that the constructed symmetric multiwavelets have the balancing property. For a filter bank related to some wavelet system, the high number of vanishing moments for wavelets
is equivalent to the same number of vanishing moment for the corresponding wavelet masks
. In practice, this means that during the analysis step of the discrete wavelet transform high frequency components of the input sequence vanish. For a multiwavelet filter bank
and
, generally, this is not the case. This property should be provided specifically, and the related notion is called the balancing property. However, in the interpolating case, balancing order for the analysis filter bank
is equal to the order of sum rule of matrix mask
. The necessary and sufficient conditions are stated in ([
41] Proposition 3.1, Theorem 4.1) (see, also, ([
16] Theorem 2.1)). The balancing property of order
n is provided for the analysis filter bank
if
, and
For the interpolating matrix mask , a vector of functions y is taken exactly as required. The last two conditions are true due to Lemma 4.
Next, we discuss several additional aspects. Let us compare the obtained results with the approach proposed in [
20] for the same problem. In [
20] the construction of matrix masks is done by transferring all required conditions into the form of systems of linear equations. The coefficients of initial matrix mask and dual matrix mask are obtained as a numerical solution of such systems. The existence of solution for any appropriate set of initial parameters is out of consideration. Wavelet matrix masks are obtained by an explicit procedure, but the study of their symmetry and approximation properties is not carried out. The approach proposed in the current paper describes the inner structure of symmetric matrix masks in terms of the modified polyphase components; coefficients of the matrix masks can be obtained even analytically, if needed. Additionally, the description of symmetry in terms of the modified polyphase components helps to study the symmetry properties of the obtained wavelet matrix masks.
In fact, the interpolating property leads to some limitations on the choice of a pair of appropriate symmetry group and dilation matrix (see condition (
18)). Additionally, the interpolating property means that the number
r of functions in refinable function vector is equal to
, and the symmetry centers are defined by digits. On the other hand, in the non-interpolating case, general matrix extension with symmetry becomes a challenging problem. In the scalar case, several methods were developed (see, for example, [
42,
43,
44] and the references therein). This is a question for future investigations, whether these methods can be modified for multiwavelets.
Finally, multivariate wavelet frames are successively used in applications related to signal processing and numerical solution of PDEs (for example, in [
5] 2D multiwavelet transform is used in the iris biometric technique for person authentication, in [
1] 2D multiwavelets are used for dimensionality reduction and features extraction in the algorithm of face recognition, in [
21] 2D multiwavelets are applied to variational image denoising and in [
4] 2D multiwavelets are used for the compression, analysis and assembly of hydrodynamic model data within Godunov-type solvers based on second-order discontinuous Galerkin methods). Mostly, 2D multiwavelets in the above-mentioned applications and in the literature are separable, i.e., they are obtained as a tensor product of 1D multiwavelets. However, there are several results that non-separable wavelets provide some advantages in certain applications compared to separable ones (see, for example, [
45,
46] for wavelet and [
47] for multiwavelets). The proposed method for the construction of symmetric non-separable multiwavelets can help to obtain multiwavelets, which potentially can lead to improvements in the mentioned and other applications. We leave these problems for future research.
5. Examples
1. Consider simple symmetry group in : and quincunx dilation matrix . The set of digits is . Clearly, condition is valid for all and . For interpolating matrix mask the row of symmetry centers is .
Consider matrix mask
defined by
Here (and below)
,
. This matrix mask is
-symmetric with respect to the row of centers
C. The matrix of symmetry centers for each matrix element is
. This matrix mask obeys the sum rule of order 2. Additionally, the eigenvalues of
are 1 and 0. The Sobolev smoothness of the corresponding refinable functions here and below can be computed using algorithm developed in [
34]. Here
,
, are at least in
See
Figure 1 for the graph of
(note that
is just a shifted copy of
). The dual matrix mask
can be taken as follows:
It is also
-symmetric with respect to the same row of centers
C and it obeys sum rule of order 1. The eigenvalues of
are 1 and 0. The corresponding refinable functions are at least in
. After the matrix extension is obtained, the wavelet matrix masks are
Note that in this case, the sets of digits associated with symmetry centers are: and . This means that each trigonometric polynomial in wavelet matrix masks is -symmetric, which implies simple symmetry of the corresponding wavelets.
2. Consider axis symmetry group in
:
Let
. The set of digits is
. Clearly, condition
is valid for all
and
. For interpolating matrix mask the row of symmetry centers is
.
Consider matrix mask
defined by
Here (and below)
,
. This matrix mask is
-symmetric with respect to the row of centers
C. The matrix of symmetry centers for each matrix elements is
. This matrix mask obeys sum rule of order 2. Also, the eigenvalues of
are 1 and 0. The corresponding refinable functions are at least in
See
Figure 2 for the graph of
(note that
is just a shifted copy of
). The dual matrix mask
can be taken as follows
It is also
-symmetric with respect to the same row of centers
C, and it obeys sum rule of order 2. Additionally, the eigenvalues of
are 1 and 0. The corresponding dual refinable functions are at least in
After the matrix extension is obtained, the wavelet matrix masks are
Note that in this case, the sets of digits associated with symmetry centers are: and . This means that each trigonometric polynomial in wavelet matrix masks is -symmetric, which implies axis symmetry of the corresponding wavelets.
3. Consider hexagonal abelian symmetry group
and dilation matrix
The set of digits are
. It can be checked that
is valid for all
and
. The row of symmetry centers for interpolating matrix mask is
.
Consider matrix mask
defined by
Here (and below) , . This matrix mask is -symmetric with respect to the row of centers C. The matrix of symmetry centers for each matrix elements is .
This matrix mask obeys the sum rule of order 2. Additionally, the eigenvalues of
are 1, 0 and 0. The corresponding refinable functions are at least in
. See
Figure 3,
Figure 4 and
Figure 5 for the graphs of
,
,
. The dual matrix mask
can be taken as follows
where
It is also
-symmetric with respect to the same row of centers
C and it obeys sum rule of order 1. Additionally, the eigenvalues of
are 1,
and 0. The corresponding dual refinable functions are at least in
Again, the matrix extension allows us to construct wavelet matrix masks. For instance, dual wavelet masks are
Analogously, primal wavelet masks can be obtained. Note that in this case the sets of digits associated with symmetry centers are:
and
,
. This means that trigonometric polynomials in columns of wavelet matrix masks are mutually symmetric, which implies mutual symmetry of the corresponding wavelets.