﻿Nonstationary Wavelets Related to the Walsh Functions

American Journal of Computational Mathematics
Vol. 2  No. 2 (2012) , Article ID: 20052 , 6 pages DOI:10.4236/ajcm.2012.22011

Nonstationary Wavelets Related to the Walsh Functions

Yuri A. Farkov, Evgeny A. Rodionov

Department of Mathematics, Russian State Geological Prospecting University, Moscow, Russia

Email: farkov@list.ru

Received March 29, 2012; revised April 25, 2012; accepted May 2, 2012

ABSTRACT

Using the Walsh-Fourier transform, we give a construction of compactly supported nonstationary dyadic wavelets on the positive half-line. The masks of these wavelets are the Walsh polynomials defined by finite sets of parameters. Application to compression of fractal functions are also discussed.

1. Introduction

As usual, let be the positive half-line, be the set of all nonnegative integers, and let be the set of all positive integers. The first examples of orthogonal wavelets on related to the Walsh functions and the corresponding wavelets on the Cantor dyadic group have been constructed in [1]; then, in [2] and [3], a multifractal structure of this wavelets is observed and conditions for wavelets to generate an unconditional basis in -spaces for all have been found. These investigations are continued in [4-10] where among other subjects the algorithms to construct orthogonal and biorthogonal wavelets associated with the generalized Walsh functions are studied. In the present paper, using the Walsh-Fourier transform, we construct nonstationary dyadic wavelets on (cf. [11-13], [14, Ch.8]).

Let us denote by the integer part of. For every, we set

where. Then

.

.

Further, we introduce the notations

where. Then the Walsh function of order is (see, e.g., [15]).

The Walsh-Fourier transform of every function that belongs to is defined by

,.

and extent to the whole space in a standard way. The intervals

, are called the dyadic intervals of range. The dyadic topology on is generated by the collection of dyadic intervals. A subset of which is compact in the dyadic topology will be called W-compact.

For any we define and by the following algorithm:

Step 1. For each choose, and, , such that

(1)

for all

(2)

with the coefficients

so that for all (cf. [15, Sect. 9.7]).

Step 3. For each put

, (3)

so that

. (4)

Step 4. Define by the formula

. (5)

Further, let us define subspaces and in as follows

,

for all.

We say that a polynomial satisfies the modified Cohen condition if there exists a W-compact subset of such that

and

. (6)

Theorem. Suppose that the masks satisfy the modified Cohen condition with a subset and there exists such that

for all,.    (7)

Then for any the following properties hold:

a) and;

b) and are orthonormal basis in and, respectively;

c),.

Moreover, we have

.

Corollary. The system

is an orthonormal basis in.

We prove this theorem in the next section. Then using the notion of an adapted multiresolution analysis suggested by Sendov [12], we discuss an application of the nonstationary dyadic wavelets to compression of the Weierstrass function and the Swartz function.

2. Proof of the Theorem

At first we prove the orthonormality of. In view of

let us show that

,.

Denote by the characteristic function of. For each we define

for Since and, for all, in some neighbourhood of zero, we obtain from Equation (3)

for all.     (8)

Let

where,. Letting, we have

that yields. By induction, we obtain

According to Equation (8), by Fatou’s lemma, we have

(9)

Consequently, and, in view of Equation (5),. It is known that if is constant on dyadic intervals of range, then (see [16, Sect. 6.2]). Therefore, each function is constant on, , which implies.

In view of Equation (7), there exists such that

for all,.

Hence, for,

.

It follows from Equation (6) that for some

for.

Since

,.

We have

.

or, taking into account Equation (3),

,

for,.

Applying the dominated convergence theorem we obtain

which means that is an orthonormal system.

Now, let us prove an orthonormality of. For any denote. Then

. (10)

Since

We have

Then from Equation (10)

,. (11)

Let us define

.

Denote. Under the unitarity of the matrices

We can write

Using the inverse Fourier-Walsh transform, we have

or,

.

With Equation (11) it yields

To conclude the proof it remains to show that

. (12)

Note, that by Equation (7) for any there exist such that and, consequently,

. (13)

For any the subspace is invariant with respect to the shift. Actually, an arbitrary can be approximated by fractions, with arbitrary large. Besides, each is invariant with respect to the shifts. By Equation (4) it is clear that.

Let. There exist such that

and hence for all. The continuity of implies that. If

, then approximating with from

and using the invariance of a norm with respect to the shift, we obtain.

Denote by the set of all such that

. By the Weiner’s theorem we can write

, for some measurable. It is clearly that and, in view of Equation (13), we have. Hence, the Equation (12) holds. The theorem is proved.

3. Numerical Experiments

For any, let,. According to [12] an adapted multiresolution analysis (AMRA) of rank in is a collection of closed subspaces, , which satisfies the following conditions:

1) for all;

2);

3) for every there is a function in with a finite support such that

is an orthonormal basis of;

4) for every there exists a filter

such that

,. (14)

The sequence from condition (4) is called a scaling sequence for given an AMRA. The corresponding a wavelet sequence can be defined by

. (15)

Denote by the orthogonal complement of in. It is known that, under some conditions, the system is an orthonormal basis of

(for more details, see, e.g., [14, Sect. 8.1]). Moreover, if denotes the projection of a function on the subset, then

and

. (16)

Let us denote

and

.

For a given array

the direct non-stationary discrete wavelet transform

maps it into

and

.

The inverse transform is defined as follows

and reconstructs by and. According to [12] in order to choose the filter to maximize in Equation (16), we must solve the following problem.

Problem 1. Let be the subset of the 2n-dimensional Euclidean space, which consists of the points satisfying the conditions

. (17)

for. Find a point for which

, (18)

where is a symmetric matrix.

Problem 1 has a solution since is a compact. But, as noted in [12], the numerical solution of this problem is not trivial even for.

Concerning the standard Haar and Daubechies (with 4 coefficients) discrete transforms see, e.g., [17]; we will denote them as SWTH and SWTD, respectively. We write NSWTH for the simplest case of a multiresolution analysis of rank 1 which is considered in [12, Sect. 3] (see also [13]). The nonstationary Daubechies discrete wavelet transform which corresponds an AMRA of rank are defined in [12] and we will use the symbol NSWTDN to denote this transform (see NSWTD1 and NSWTD2 in the tables below).

Method A associated with one of the mentioned above discrete wavelet transforms (cf. [17, Chap.7]) consists of the following steps:

Step 1. Apply the discrete wavelet transform times to an input array and get the sequence

.

Step 2. Allocate a certain percentage of the wavelet coefficients with lagest absolute value (we choose 10%) and nullify the remaining coefficients.

Step 3. Apply the inverse wavelet transform to the modified arrays of the wavelet coefficients.

Step 4. Calculate, where is a reconstructed array.

In Method B the second step is replaced on the uniform quantization and the forth step is replaced on the calculation of the entropy of a vector, obtained in the third step.

We recall that is a vector uniform quantization for given vector, if

where is the length of the quantization interval.

The value will be calculated by

The Shannon entropy of is defined by the formula

where is frequency of the value.

Let us consider a similar approach associated with the following problem:

Problem 2. Let. Denote by the set of all points such that

For every we define

for. Find a point for which

(19)

where is a symmetric matrix.

Given an array, we define the matrix in Problem 1 and Problem 2 by

and

respectively. Here for. Suppose that is a solution of Equation (19). Then the direct and inverse nonstationary discrete dyadic wavelet transforms are defined by

, ,

where and. We

Table 1. Values of the square error corresponding to Method A.

Table 2. Values of the entropy obtained by Method B.

denote these discrete transforms as NSWTL1 if and as NSWTL2 if.

Let us recall that the Weierstrass function is defined as

and the Swartz function is defined as

where. We will consider arrays with elements a8,k =or a8,k =,. Then we use the Matlab function fminsearch to solve the optimization problems in Equations (18) and (19). The results of these numerical experiments are presented in Tables 1 and 2. We see that in several cases the introduced nonstationary dyadic wavelets have an advantage over the classical Haar and Daubechies wavelets.

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