Abstract
In this paper, we consider the dual wavelet frames in both continuum setting, i.e., on manifolds, and discrete setting, i.e., on graphs. Firstly, we give sufficient conditions for the existence of dual wavelet frames on manifolds by their corresponding masks. Then, we present the formula of the decomposition and reconstruction for the dual wavelet frame transforms on graphs. Finally, we give a numerical example to illustrate the validity of the dual wavelet frame transformation applied to the graph data.
1. Introduction
Interest in signal processing algorithms in various applications has increased in recent years. Examples of these signal processing algorithms include sensor networks, transportation networks, the Internet, and social networks. In these applications, data are defined on topologically complicated domains, such as high-dimensional structures, irregularly sampled spaces, and manifolds. Such datasets are commonly referred to as big data. Several attempts have been made to extend conventional signal processing techniques to big data. As described in [1], the success of wavelet frames for data defined on flat domains has encouraged research on the generalization of wavelets and wavelet frames on topologically complicated domains. For example, Coifman and Maggioni constructed diffusion wavelets and diffusion polynomial frames on manifolds in [2, 3]. Geller and Mayeli studied the construction of wavelets on compact differentiable manifolds in [4]. Hou et al. constructed the well-known Mexican hat wavelet on a manifold geometry in [5]. Graphs are effective ways to represent the geometric structures of data on complicated domains. Hammond et al. first used spectral graph theory to characterize spectral graph wavelets on graphs in [6]. Leonardi and Van De Ville introduced Meyer-like wavelets and scaling kernels that result in tight spectral graph wavelet frames in [7, 8]. Shuman et al. first characterized a family of systems of uniformly translated kernels in the graph spectral domain that gave rise to tight frames of atoms generated via generalized translation on a graph in [9]. In addition, wavelet frames can be applied to sparse representation of piecewise smooth functions. Dong developed a system frame for sparse representation on graphs using tight wavelet frames and applications in [1]. Wang and Zhuang provided a complete characterization for the tightness of framelet systems in both continuous and discrete framelets on a manifold in [10]. Analogous to the standard Cohen-Daubechies-Feauveau (CDF) construction of factorizing a maximally flat Daubechies half-band filter, Narang and Ortega proposed the design of graph-QMF for arbitrary undirected weighted graphs in [11]. Tanaka and Sakiyama studied -channel oversampled filter banks satisfying the perfect reconstruction condition for graph signals and showed oversampled graph filter banks for applications to graph signal denoising in [12].
As a generalization of tight wavelet frames, pairs of dual wavelet frames have proved particularly useful in signal denoising and many other applications where translation invariance or redundancy is important. Han first characterized homogeneous dual wavelet frames for any general dilation matrix in the space in [13]. In this case, , where denotes a measurable subset of . Chui et al. were the first to completely characterize tight dual frames with maximum vanishing moments generated using two generators derived from the same refinable function in [14]. Daubechies and Han constructed pairs of dual wavelet frames from two refinable functions in [15]. Ehler constructed compactly supported multivariate pairs of dual wavelet frames called biframes, based on the mixed oblique extension principle in [16]. Han and Shen generalized the mixed extension principle in to obtain and characterize dual Riesz bases in Sobolev spaces in [17]. Jiang and Pounds studied sixfold symmetric biframes with four framelets (frame generators) for triangular-mesh-based surface multiresolution processing in [18]. Readers can refer to [19–22] for more construction methods and properties regarding dual wavelet frameworks in .
Motivated by these and other applications, in this paper, we introduce and study dual wavelet frames on manifolds. We consider the characteristics for the existence of dual wavelet frames in both continuous and discrete settings. In particular, discrete dual wavelet frame transforms and numerical examples on graphs are provided. The results may serve as an analysis tool for the processing of graph data. The paper is organized as follows. In Section 2, we briefly review some basic notations and concepts related to our present work. In Section 3, the conditions for the existence of dual wavelet frames on manifolds are proved. In Section 4, we discuss discrete dual wavelet frame transforms on graphs (DWFTG) and provide a multiresolution analysis (MRA) on a graph. For computation and applications in practice, in Section 5, we show that discrete dual wavelet frame transforms on graphs can be achieved by using low-degree Chebyshev polynomials. In addition, we give numerical simulations of fast dual wavelet frame transforms on graphs to demonstrate the efficient implementation of the framelet transforms.
The following is the list of notations used in this paper. Let , , and denote the set of real numbers and the set of natural number and integers, respectively. Let denote the conjugate of complex number . Let be the set of 2 convergent sequences on . Let be a manifold. For a given compact, connected, and smooth Riemannian manifold , let be a space of complex-valued square integrable functions on . Let be the Laplace-Beltrami operator on with respect to the metric . is the Kronecker delta function with if and 0 otherwise. Moreover, let denote a undirected, connected, weighted graphs, where is a discretization of a given manifold , is an edge set, and denotes a weight function. We consider here only finite graphs with .
2. Definitions and Concepts on Manifolds
Throughout this paper, we will use the following notations for the inner product and norm for the space of .where , (or simply ) denotes the measure from the area element of a Riemannian metric on , a probability measure satisfies , and endowed with the norm.
Definition 1. Let be the Laplace-Beltrami operator on , and ; if the eigenvalue problem holds, then the sequence of pairs is called a well-defined eigensystem for .
Definition 2. The two sequences pairs and are said to be a biorthonormal eigenpair for , if they satisfy the following properties:
(1) A pair of sequences form an biorthonormal basis of ; i.e., .
(2) Each of the two sequences is a nondecreasing sequence of nonnegative numbers and satisfies and , respectively.
Since and are an biorthonormal eigenpair for , the following can be given.
Definition 3. The generalized Fourier transform of a function is defined by with , .
So, we can get the following result [6].
Theorem 4. Any has the Fourier expansion in and the Parseval relation holds. In particular, when , Parseval’s identity holds.
Let refinable function and its dual function which satisfy the following scaling equation: where , are all finitely supported and called a refinement mask or the low filter, respectively.
Given such a pair refinable function and with mask and , we define a pair functions set and as follows: where and are called wavelet frame masks or the high pass filters of the system, respectively.
For and defined by (7), denote and . So, there is the following definition of a quasi-affine system on manifold [1].
Definition 5. Let be a set of functions. A quasi-affine system is defined as where
In fact, can be explained as the dilation and translation of at scale and a point . And again, a similar definition for can be given as follows: where
Definition 6. Given a Hilbert space and a set of vectors , if there exist two positive constants and such that the inequality holds for all , then a set of vectors is called a frame in .
Now, we begin to give the definition of a pair of dual wavelet frame on manifolds.
Definition 7. For given and , if
and defined by (8) and (10) are in
each of and is a wavelet frame in
in -sense, the following perfect reconstruction formula holds for any or, equivalently, Then the pair of systems is called a dual wavelet frame for . In particular, when , the frame is said to be a tight frame for .
3. Existences of Dual Wavelet Frames on Manifolds
In this section, we give sufficient conditions for the existence of dual wavelet frames for by their corresponding masks. First, we give Weyl’s asymptotic formula [23, 24] and Griesers uniform bound of the eigenfunctions [25] as below similar to the way of literature [1, 10].
Lemma 8. Let and be well-defined eigensystem of the Laplace-Beltrami operator on ; then and where the constant depends only on the dimension of the manifold. Here, the symbol means that there exist positive constants , independent of such that .
Lemma 9. Suppose a pair of functions in and the corresponding pair of masks satisfying equations (6) and (7). Let and , be defined as in (9) and (11), respectively. If then, for any ,
Similar to the literature [1], we can obtain the following results.
Theorem 10. Let in be a pair functions with masks pair satisfying (6) and (7). Assume that the refinable function pair satisfy conditions (17) and corresponding masks satisfy for near the origin. Then, the pair of systems given by (8) and (10) forms a dual wavelet frame in provided the equality holds for .
Proof. For simplicity’s sake, we define the operation in as In particular, By formulas (9) and (11), we obtain By Parseval’s relation, we have Further, Therefore, we have Thus, By , we get Therefore, and hence To obtain (13), we first need to prove that Due to By assumption (19), we have , ; thus, for each , and hence and are bounded. Therefore, , and then Second, we show that Since and , for , we have Therefore, by the boundedness of and , we have .
This completes the proof of the theorem.
4. Discrete Dual Wavelet Frame Transforms on Graphs
Discrete dual wavelet frames on are more desirable in practice. Graphs are usually understood as a certain discretization or a random sample from some smooth Riemannian manifold. In this section, we discuss discrete dual wavelet frame transforms on graphs.
For a given undirected, connected, weighted graphs ; let be a adjacency matrix for a weighted graph with entries , where
Let represent the degree matrix defined as follows: where is called the degree of a vertex .
The graph Laplacian plays an important role in the analysis and processing of graph data. The consistency of the graph Laplacian to the Laplace-Beltrami operator was studied in [26–28].
Definition 11. A nonnormalized graph Laplacian is defined as follows:
Let be the set of pairs of eigenvalues and eigenfunctions of ; then,
without losing generality, we have ;
the eigenfunctions form an orthonormal basis for all functions on the graph; that is, .
Definition 12. Let be a function on the graph . Then its Fourier transform is defined by
Suppose is sampled from the underlying function by , where the dilation scale is the smallest integer such that Therefore, its Fourier transform is given by Note that the scale is selected such that for .
Given a graph function , we define the discrete level dual wavelet frame decomposition operator as where is its dual wavelet frame coefficient. Let and where is the reconstruction operator.
Now, we can define the discrete -level dual wavelet frame decomposition and reconstruction algorithm for graph data as shown below.
Algorithm 13. Given a signal , , and the associated trigonometric polynomials and , we assign . Then, the discrete -level fast dual wavelet frame decomposition and reconstruction for is given as follows (in the Fourier domain). (1) Decomposition: for each .(a)Obtain the low-frequency approximation of at level : (b)Obtain the dual framelet coefficients of at level : (2) Reconstruction: for each ,
Then we have the following perfect reconstruction formula from to through .
Theorem 14. If a given set of masks and in satisfy (20), then, the discrete dual wavelet frame transforms and defined on satisfy
Proof. For simplicity, we only give proof in the case of L=2 . For a general , it can be obtained in a similar way. For , we have with and . In accordance with (45) and (46), we have Therefore, by assuming in (47), we obtain By assuming in (47), we obtain This shows that .
5. Polynomial Approximation and Numerical Simulations
5.1. Polynomial Approximation and Fast DWFTG
Chebyshev polynomials have an important application in the approximation theory. In this section, we describe the details of the dual wavelet frame transform on graphs based on polynomial approximation by using the method proposed in literature [1, 6]. One of the advantages of using lower Chebyshev polynomials to approach trigonometric polynomials is the efficiency of computing if the Laplacian is sparse.
We recall the relevant definitions and properties of the Chebyshev polynomials.
Definition 15. The Chebyshev polynomial is defined as follows: , .
Proposition 16. The Chebyshev polynomials have the following properties.
and are orthogonal with respect to the weight function in the interval ; that is, where .
satisfies recursive relations For a given with , it has a convergent Chebyshev series where the Chebyshev coefficients is .
Thus, a smooth function has the following accurate approximate: and we denote the Chebyshev polynomial approximation of and as
The Laplacian is a symmetric matrix so that the eigenvalue decomposition exists, where and columns of are the eigenvectors. Now, we can use the above approximation to speed up the decomposition transform. First, we can rewrite the decomposition transform in the following matrix form: where .
Due to , we have However, where ; i.e., , and where ; i.e., .
Therefore, (58) can be equivalently written in physical domain:
Substitute (56) into (62) and notice that are polynomials; we obtain the fast DWFTG as follows:
Next, reconstruction transform can be obtained similarly; i.e., (47) can be rewritten as the following matrix form: Therefore, the above formula can be equivalently written in physical domain: We can obtain the fast dual wavelet frame reconstruction transform on graphs as follows: And we have .
5.2. Numerical Simulations of DWFTG
In this section, we give a numerical example to illustrate the validity of the dual wavelet frame transformation applied to the graph data.
Example 17 (linear). Let be the refinement masks of the continuous linear B-splines and supported on . Define Then, this group of masks satisfy (20). Hence, the system pair defined in (8) and (10) associated with the mask , is a dual wavelet frame for .
Example 18 (cubic). Let be the refinement masks of the continuous linear B-splines supported on and be the refinement masks of the cubic B-splines supported on . Define Then, this group of masks satisfy (20). Hence, the system pair defined in (8) and (10) associated with the masks , is a dual wavelet frame for .
Example 19 (pseudo). Let and be the mask of the m order pseudo-spline. Define Then, this group of masks satisfy (20). Hence, the system pair defined in (8) and (10) associated with the masks , is a dual wavelet frame for .
In our simulations, we choose the dual wavelet frame system given in Example 17. We accurately approximate the trigonometric polynomial masks and in Example 17 by low-degree Chebyshev polynomials; see Table 1.
In numerical experiment, we take points sampled on the manifold and we build a weighted graph by setting edge weights for . For larger datasets, this graph could be sparsified by thresholding the edge weights. We use and threshold the adjacency matrix to limit the number of nearest neighbors of each vertex to for computing the underlying weighted graph. The functions are generated by mapping two images, Slope and Eric, onto the graph ; see Figure 2 of the literature [1]. We perform levels of DWFTG and use for the Chebyshev polynomial approximation of the masks.
As described in Algorithm 13, we can decompose graph data on Slope and Eric to obtain the dual wavelet frame coefficients for and by the decomposition of (63) (see Figures 1 and 2) and reconstruct graph data on Slope and Eric from these framelet coefficient sequences by the reconstruction of (66); see Figure 3.
The graphs in Figures 1 and 2 show that the dual wavelet frame systems provide both good data approximation and detail separation based on multiresolution analysis; thus we demonstrate the ability of the fast dual wavelet frame transform for graph data analysis.
The computation time and reconstruction errors of a -level decomposition and reconstruction of the DWFTG are summarized in Table 2. For a similar computation time, the reconstruction error of DWFTG using linear masks is smaller than DWFTG using cubic masks and DWFTG using pseudo masks. However, the DWFTG using linear masks is a more redundant transformation than the DWFTG using cubic masks and DWFTG using pseudo masks.
Remark 20. Note that, the sparsity of the high frequency coefficients is related to the support set of the scale function and the symmetry of the filter in image processing. When the support set of the scale function is smaller and the filter banks pair generating the dual wavelet frame system are symmetrical, the high frequency coefficients are good sparse. On the contrary, the sparsity of the image is not guaranteed. At the same time, the reconstruction error will be slightly larger with the number of filters increasing. But the result is also ideal.
6. Conclusion
Wavelet frames on manifolds are a new research topic in the area of wavelet analysis. This paper is primarily concerned with the construction of the dual wavelet frames on manifolds and graphs using the machinery of extension principles. We have obtained the characterizations for the existence of dual wavelet frames in both the continuous and the discrete setting. In addition, we have also obtained the decomposition and reconstruction for the dual wavelet frame transforms on graphs. A complete section has been devoted to studying discrete dual wavelet frames on graphs using low-degree Chebyshev polynomials. Finally, some numerical simulations of fast dual wavelet frame transform on graphs have also been carried out to demonstrate the efficiency and accuracy of the framelet transforms.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors appreciate the support of the National Natural Science Foundation of China (Grant no. 21676010). We would like to thank the authors and experts whose literatures are cited in this paper and those who have taken the time to review our paper.