International Journal of Computational Mathematics
Volume 2014 (2014), Article ID 531562, 7 pages
http://dx.doi.org/10.1155/2014/531562

Research Article

Multiresolution Analysis Based on Coalescence Hidden-Variable Fractal Interpolation Functions

G. P. Kapoor¹ and Srijanani Anurag Prasad²

¹Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Uttar Pradesh 208016, India
²Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Indore By-Pass Road, Bhauri, Bhopal, Madhya Pradesh 462066, India

Received 23 June 2014; Accepted 20 November 2014; Published 11 December 2014

Academic Editor: Neeraj Mittal

Copyright © 2014 G. P. Kapoor and Srijanani Anurag Prasad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Multiresolution analysis arising from Coalescence Hidden-variable Fractal Interpolation Functions (CHFIFs) is developed. The availability of a larger set of free variables and constrained variables with CHFIF in multiresolution analysis based on CHFIFs provides more control in reconstruction of functions in than that provided by multiresolution analysis based only on Affine Fractal Interpolation Functions (AFIFs). Our approach consists of introduction of the vector space of CHFIFs, determination of its dimension and construction of Riesz bases of vector subspaces , consisting of certain CHFIFs in .

1. Introduction

The theory of multiresolution analysis provides a powerful method to construct wavelets having far reaching applications in analyzing signals and images [1, 2]. They permit efficient representation of functions at multiple levels of detail; that is, a function , the space of real valued functions satisfying , could be written as limit of successive approximations, each of which is smoothed version of . The multiresolution analysis was first introduced by Mallat [3] and Meyer [4] using a single function. The multiresolution analysis based upon several functions was developed in [5–7]. In [8], multiresolution analysis of was generated from certain classes of Affine Fractal Interpolation Functions (AFIFs). Such results were then generalized to several dimensions in [9, 10]. In [11], orthonormal basis for the vector space of AFIFs was explicitly constructed. A few years later, Donovan et al. [12] constructed orthogonal compactly supported continuous wavelets using multiresolution analysis arising from AFIFs. Bouboulis [13] generated multiresolution analysis of using AFIF on and constructed multiwavelets which are orthonormal but discontinuous. The interrelations among AFIFs, multiresolution analysis and wavelets are treated in [14]. In [15], multiresolution analysis is developed for a Hilbert space constructed from Hausdorff measures , on and, in particuluar, on a Cantor set using linear contraction. This development was later improved in [16] by employing a nonlinear fractal system to construct wavelets with Fourier basis with respect to some fractal measure. The details on implementation of recent work on multiresolution analysis with AFIF bases can be found in [17]. It is desirable [18] that the wavelet function should reflect the features present in the original function but AFIF based wavelets generally cannot exhibit satisfactorily the features of functions simulating natural objects or outcome of scientific experiments that are partly self-affine and partly non-self-affine. The Coalescence Hidden-variable Fractal Interpolation Functions (CHFIFs) introduced in [19] are ideally suited for such purposes. However, multiresolution analysis of based on CHFIFs has hitherto remained unexplored. In the present work, such a multiresolution analysis using CHFIFs as basis functions is developed. The availability of a larger set of free variables and constrained variables in our multiresolution analysis based on CHFIFs additionally provides more control in reconstruction of functions in than that provided by multiresolution analysis based only on affine FIFs.

The organization of the paper is as follows. The construction of a CHFIF is briefly summarized in Section 2. The vector space of CHFIFs is introduced in Section 3 and a few auxiliary results, including a result on determination of dimension of this vector space, are found in this section. In Section 4, first Riesz bases of vector subspaces , consisting of certain CHFIFs in are constructed. The multiresolution analysis of is then carried out in this section in terms of nested sequences of vector subspaces .

2. Construction of a CHFIF

In this section, a brief introduction on the construction of CHFIF is given. A Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is constructed as the graph of the attractor of a suitably defined Iterated Function System (IFS).

Given an interpolation data , where , a CHFIF is constructed as follows. Consider a generalized interpolation data , where are real numbers. We denote the interval by and the intervals by for . Define the functions and by

where , , and the functions satisfy the join-up conditions:

In (2), the variables , are free variables, are constrained variables such that , , , and , are linear polynomials given by

The desired IFS for construction of a CHFIF for the generalized interpolation data is now defined as

where

The following theorem gives the existence of an attractor of the IFS defined by (5) associated with the generalized interpolation data.

Theorem 1 (see [19]). Let be the IFS defined by (5) associated with the generalized data . Let , and in the definition of satisfy , and for . Then there exists a metric on , equivalent to the Euclidean metric, such that the IFS is hyperbolic with respect to . In particular, there exists a unique nonempty compact set such that .

The following theorem is instrumental for precise definition of a CHFIS.

Theorem 2 (see [19]). Let be the attractor of the IFS for the given interpolation data. Then G is graph of a continuous function such that for , that is, .

Suppose is written component-wise as . Then the Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is defined as follows.

Definition 3. The Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) for the given interpolation data is defined as the first component of the function .

It is easily seen that the graph of CHFIF is the projection of the graph of on .

3. Auxiliary Results

In order to develop the multiresolution analysis of based on CHFIF, the space of CHFIF is introduced in this section. Further, a few auxiliary results and a result on the dimension of the vector space of CHFIF are found here.

Let

and , where and are polynomials of degree at most given by (4). Then, is a vector space with usual pointwise addition and scalar multiplication, where is the class of polynomials of degree at most . It is easily seen that on , the set of bounded functions from to with respect to maximum metric , the function defined by

for , , is a contraction map. Therefore, by Banach contraction mapping theorem, has a unique fixed point . By join-up conditions (3), it follows that , the set of continuous functions from to . The following proposition gives the existence of a linear isomorphism between the vector space and the vector subspace of defined by .

Proposition 4. The mapping defined by is a linear isomorphism.

Proof. The assertion of the proposition is proved by establishing
(i) , , where and are written component-wise as and , (ii) , (iii) is onto and (iv) is one-one.
The identity (i) follows by equating the components of left and right hand side in the identity .
(ii) : By the definition of :

Using identity (i), it follows that

The above equation gives the following on simplification:

Therefore, is a fixed point of for all and . By uniqueness of fixed point of , it follows that .
(iii) is onto: Let . Define and for . Suppose , where . Then whenever . Also .
(iv) is one-one: Let for all values of . Then, for every .

To introduce the space of CHFIFs, let the set consisting of functions be defined as is  a  CHFIF  passing  through , and  is  an  AFIF  passing  through  . Then, is a vector space, with usual pointwise addition and scalar multiplication. The desired space of CHFIFs is now defined as follows.

Definition 5. Let be the set of functions that are first components of functions . The space of CHFIFs is the set together with the maximum metric .

It is easily seen that the space of CHFIFs is also a vector space with pointwise addition and scalar multiplication. The following proposition gives the dimension of .

Proposition 6. The dimension of space of CHFIFs is .

Proof. Consider the operator . The operators , , where is the set of bounded functions, satisfy

for . By Proposition 4 and (13), it follows that is completely determined by for . Further, it follows by (12) that depends on . Then, for , the function is the unique CHFIF passing through , while the function is the unique AFIF passing through . Hence,

Now, consider the projection map . Then, kernel of is a proper subset of and consists of elements of the forms and . For the element , it is observed that for . Hence, for all , it is seen that . With , it follows that and , . Consequently, if then is a linear polynomial. So, dimension of . Therefore, by Rank-Nullity Theorem, .

Remark 7. By Proposition 4 and (14), it follows that the map defined by is a linear isomorphism, where , is the unique CHFIF passing through the points , and is the unique AFIF passing through the points , , and . Thus, is linearly isomorphic to . Consider the metric space , where is given by , , , , and . Then, with the restriction of metric on the set , it is observed by (8) that the maps and are continuous. Hence is closed and complete subspace of .

Remark 8. Let be a sequence in such that and be a convergent sequence in , where the functions are AFIFs. Since is closed, . Thus, and, consequently, is closed subspace of .

4. Multiresolution Analysis Based on CHFIF

In this section, the multiresolution analysis of is generated by using a finite set of CHFIFs. For this purpose, the sets , consisting of a collection of CHFIFs, are defined and it is shown that these form a nested sequence. The multiresolution analysis of is then generated by constructing Riesz bases of vector subspaces consisting of certain orthogonal functions in .

To introduce certain sets of CHFIFs needed for multiresolution of , let be a collection of functions such that and , and be a collection of functions such that and , , the set of all real valued continuous functions defined on which vanish at infinity. Define the set as

where , .

That the set is not empty is easily seen by considering a function , with and for , which obviously belongs to . Let, for ,

The sets and are seen to be closed sets as follows. Let be a sequence in such that . Now, . By Remarks 7 and 8, it is observed that is a CHFIF and is an AFIF, . Thus, , which implies that is a closed set. Consequently, and are closed sets. Now, for , define

It follows from Proposition 4 that the sets , with -norm, are vector subspaces of . The following proposition shows that these subspaces form a nested sequence.

Proposition 9. The subspaces , form a nested sequence .

Proof. To show that for all , it suffices to prove the inclusion relation for . Let . Then, for some . If then, implies, for , , where for all , . Expressing and in matrix form as and , it is observed that nonzero entries in matrices and occur at the same places. Consequently, is graph of , so that . It therefore follows that is a CHFIF on the interval . Thus, the function is a CHFIF on the interval and consequently, .

In order to generate a multiresolution analysis of using CHFIFs, the inner product on the vector space , is defined by . Using and , it is observed that, for ,

where , and , ; and , , are given by (1), (2), and (4), respectively, for the interpolation data and . Using (18), the set of orthogonal functions that forms the Riesz basis of set is now constructed as follows.

Let the free variables , and constrained variables , , , in the construction of CHFIF be chosen such that for at least one . Consider, the points and , , given by

and a set of functions , , where the CHFIF passes through the points , , being the th component of and AFIF passes through the points , , being the th component of . Let the function , , be the extension of the function such that for and for .

For ensuring the orthogonality of the functions with respect to the inner product in , let the values of , and , , in (19) be chosen such that

Let, for ,

The free variables and constrained variables , , in (2) are variables and , is a system of equations. Suppose there exist no and , , in such that , ; then dimension of , which is a contradiction. Hence, there exists at least one set of and , , in such that , . The free variables and constrained variables , , in (2) are chosen such that, for , , and .

It is easily seen that the functions , , , and the functions , , are linearly independent. Now, by (8), and , , where and are linear polynomials. By (20), the functions , , are nonlinear polynomials. Hence, if and only if , which implies , , are linearly independent. The linear independence of , together with (21) now ensures the same number of orthogonal functions by applying the Gram-Schmidt process.

Let , , be a sequence of orthogonal functions obtained from the sequence , by the Gram-Schmidt process. Set

It is easily seen by Proposition 6 that none of the functions , , are identically zero. Further, by (21) and (22), it follows that is an orthogonal set. This is the set that leads to the generation of multiresolution analysis of in the following theorem.

Theorem 10. Let free variables , and constrained variables , , , in the construction of CHFIF be chosen such that for at least one and let , given by (22) be such that , . Then,

where . Also, the set generates a continuous, compactly supported multiresolution analysis of .

Proof. It is obvious that functions , , are compactly supported and are elements of . Now, implies is a CHFIF for some . Since every is determined by and , , the function has a unique expansion in terms of the functions , , and their integer translates. Hence, the function has a unique expansion in terms of the functions , and their integer translates. Thus, CHFIF has a unique expansion in terms of the functions , , and their integer translates, that is, where . Since is arbitrary, . Let be a sequence in such that and let be a convergent sequence in , where are AFIFs. Since is closed, which gives that is a CHFIF. Hence, . It therefore follows that is closed and .
Now, the following steps show that the set indeed generates a continuous, compactly supported multiresolution analysis of .
(a) By Proposition 9, it follows that .
(b) To prove that , let , and where if and if . Since the space is finite dimensional over , the norms and restricted to are equivalent. Hence, there exists a positive constant such that for all . By the property of translation invariance, it is observed that for any . Thus, for any . It therefore follows by the definition of that for all . Consequently, if , then which implies .
(c) For showing that , let , where is a CHFIF passing through and is an AFIF passing through . For all , by (19),

where

Now, since are continuous and compactly supported, by using (b) and Proposition 3.1 of [11], it follows that is dense in .
(d) For proving that the functions , , and their integer translates form a Riesz basis for , let be the smallest eigenvalue of the matrix

Since the functions and are linearly independent, the determinant of the matrix is positive which implies . Taking and , it is seen that, for every , . Further, the functions , , and their integer translates are mutually orthogonal. Therefore, the functions , , and their integer translates form a Riesz basis for .

Remark 11. By Theorem 10, it follows that the set , where , , actually generates a continuous, compactly supported multiresolution analysis of by orthonormal functions.

5. Conclusions

In this paper, multiresolution analysis arising from Coalescence Hidden-variable Fractal Interpolation Functions is developed, since CHFIF based wavelets generally more satisfactorily preserve the features of the functions simulating natural objects or outcome of scientific experiments that are partly self-affine and partly non-self-affine compared to AFIF based wavelets. The availability of a larger set of free variables and constrained variables with CHFIF in multiresolution analysis based on CHFIFs provides more control in reconstruction of functions in than that provided by multiresolution analysis based only on affine FIFs. In our approach, the vector space of CHFIFs is introduced, its dimension is determined, and Riesz bases of vector subspaces , consisting of certain CHFIFs in are constructed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The Srijanani Anurag Prasad would like to thank CSIR for Research Grant no. 9/92(417)/2005-EMR-I for the present work.

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