Introduction - EXTENDING CLASSICAL MULTIRATE SIGNAL PROCESSING THEORY TO GRAPHS

EXTENDING CLASSICAL MULTIRATE SIGNAL PROCESSING THEORY TO GRAPHS

6.1 Introduction

Multirate analysis for graph signals has been of interest since the introduction of the field of graph signal processing. The first set of papers, pioneered by Narang and Ortega [129, 127, 125, 4, 66, 160] showed how two-channel filter banks can be constructed on graphs, and went on to develop elegant techniques for the design of down-sampled, two-channel perfect reconstruction filter banks on bipartitie graphs.

These results were developed for graphs that have a real, symmetric adjacency matrix, and all results were based on the graph Laplacian. Studies in [57, 56, 58] mainly focus on circulant graphs and analyze two-channel decomposition of graph signals. Multirate decomposition can be achieved by iterative application of 2-channel systems. The study in [192] proposes to combine decimators and filters for construction of a filter bank on a graph. Motivated by Haar filter banks in the classical theory, a graph filter bank is developed using the partitions of the graph.

Inspired by the pioneering contributions of [152] and [129], this chapter extends many of the basic concepts of classical multirate signal processing and filter bank theory to graphs. We first develop the equivalent of fundamental ideas such as noble identities, aliasing, and polyphase decompositions in graph multirate systems.

Then, a detailed general theory for𝑀-channel filter banks is developed. The graphs are assumed to be very general as in [152], with a possibly non-symmetric and complex adjacency matrix.

In the context of graph signal processing a linear filter is just a square matrix. By a cascade of such matrices one can trivially construct a graph filter bank. Problems with this approach and reasons why we focus on polynomial filters are detailed in in Section 6.3. We will see in this chapter that the extension of classical multirate signal processing theory to graphs is nontrivial, and requires certain mathematical restrictions on the graph adjacency matrixA. While some of the results of classical filter bank theory extend easily, some of the deeper results unfold a lot of surprises – some extend and some do not extend to the case of graphs. For example, the classical noble identities [200] cannot be taken for granted, and require some restrictions on

the graph matrix A. Similarly, one cannot take it for granted that the delay chain system [200] is a perfect reconstruction filter bank (an easily proved result in the case of classical filter banks). It will also be shown that 𝑀-partite extensions of the bipartite results in [129] will not in general work for 𝑀 channel filter banks, but a more restrictive condition called𝑀-block cyclic property should be imposed on the graph. While a number of results in this chapter require this property, there are ways to relax it as explained in Section 6.15, and also in specific theorem statements.

While dealing with graphs, we often make comparisons with conventional multirate systems and filter banks defined in the time domain [200, 199, 204, 164, 43, 79].

On rare occasions we also make comparisons with systems defined in the cyclic (periodic) time domain that is equivalent to a graph with a specific cyclic adjacency matrix (Eq. (12) in [153]). These systems defined in the time-domain (or cyclical time domain on occasions) will be referred to as “classical” systems, “classical”

filter banks, and so forth.

6.1.1 Scope and Outline of This Chapter

After introducing the canonical downsampling and upsampling operators on graphs, we begin with a study of noble identities. These identities are known to be important in theoretical developments and practical implementations of classical multirate systems [200, 43]. For the case of graphs we will show in Section 6.2.2 that the noble identities make sense only for graphs with a certain specific structure on the adjacency matrix (Theorems 6.1 and 6.2). We then show in Section 6.2.3 that the delay chain filter bank (or the lazy filter bank) does not in general have perfect reconstruction property for arbitrary graphs. We introduce Type-1 and Type-2 polyphase representation of polynomial filters in Section 6.2.4. Section 6.3 discusses how one can trivially construct a graph filter bank, and motivates the use of polynomial filters. In order to extend the results for bipartite graphs on 2- channel systems to𝑀-channels, one may propose to use𝑀-partite graphs rather than bipartite graphs. In Section 6.4 we briefly discuss that such a generalization is not useful. Section 6.5 introduces𝑀-block cyclic graphs that are important for many of the later developments in this chapter. The eigenstructure of𝑀-block cyclic graphs, which forms the foundation for many of these results, is developed in Section 6.6 (Theorem 6.9). Many of the results developed in this thesis are therefore valid only for graphs that satisfy either the𝑀-block cyclic property or the eigenvector structure of 𝑀-block cyclic graphs. In Section 6.15 we also discuss how this restriction can be removed, and what the price paid is.

The concepts of spectrum folding and aliasing are developed in Section 6.8 for graphs that have an eigenvector structure similar to those of𝑀-block cyclic graphs.

These will be used later to develop perfect reconstruction filter banks.

Section 6.9 embarks on a study of three related properties of linear systems on graphs: namely the polynomial property, the shift invariance property, and the so- called alias-free property. While these properties are identical in classical signal processing theory, such is not the case on graphs. Some of these interrelations were developed in [152], but Section 6.9 goes deeper and establishes the complete picture. This will be useful for obtaining a deeper understanding of alias-free maximally decimated 𝑀-channel graph filter banks to be studied in this chapter.

With the graph adjacency matrix regarded as a shift operator [152, 153], it will be shown in Section 6.10.1 that the graph filter bank is a shift-variant system, although it is in general notperiodicallyshift-variant as in classical time domain filter banks.

We then establish the conditions on the adjacency matrix A for the periodically shift-varying property and show that it is exactly identical to the conditions for the existence of graph noble identities (Theorem 6.17).

Then in Section 6.11 we consider graphs that satisfy the specific eigenvector structure (i.e.,𝛀-graphs). These graphs are more general than𝑀-block cyclic graphs, which satisfy both the eigenvalue and eigenvector conditions. For such graphs we define band-limited graph signals and polynomial perfect interpolation filters for decimated versions of such signals. This allows us to develop a class of perfect reconstruction filter banks for𝛀-graphs (Theorem 6.19), similar to ideal brickwall filter banks of classical sub-band coding theory. Such graph filter bank designs are usually not practical because the polynomial filters have order 𝑁-1 (where the graph has 𝑁 vertices and 𝑁 can be very large). Furthermore these specific filters for perfect reconstruction are very sensitive to our knowledge of the graph eigenvalues.

In Section 6.12 we develop graph filter banks on 𝑀-block cyclic graphs (defined and studied in Section 6.5). For such graphs the eigenvalues and eigenvectors are both constrained as in (6.56), (6.57). We show that for such graphs the condition for perfect reconstruction is very similar to the PR condition in classical filter banks (Theorem 6.21). For such graphs, it is therefore possible to design PR filter banks by starting from any classical PR system. In particular it is possible to obtain PR systems with arbitrarily small orders (independent of the size of the graph) for the polynomial filters. Furthermore the PR solutions{𝐻_𝑘(A), 𝐹_𝑘(A)}are not sensitive to graph eigenvalues.

In Section 6.13 we consider polyphase representations for graph filter banks. This is useful to obtain alternative theoretical representations, as well as implementations.

Unlike classical filter banks, these polyphase representations are not always valid.

They are valid only for those graphs that satisfy the noble identity requirements (Theorem 6.3). For such graphs, the PR condition and the alias cancellation condition can further be expressed in terms of the analysis and synthesis polyphase matrices if the graphs also satisfy the 𝑀-block cyclic property (Theorems 6.25 and 6.26). Interestingly these alias cancellation conditions are somewhat similar to the pseudo-circulant property developed for classical filter banks in [200].

In Section 6.14 we consider frequency responses of graph filter banks inspired by similar ideas in [153, 160]. We will see that this concept can be meaningfully developed for filter banks on𝑀-cyclic graphs with all eigenvalues on the unit circle, but not for arbitrary graphs.

Finally in Section 6.15 we show that the eigenvector structure in (6.57) (𝛀-structure) can be relaxed simply by considering a transformed graph based on similarity transformations. This generalization therefore extends many of the results in this chapter to more general graphs. In short, all results that we developed for𝛀-graphs (e.g., Theorems 6.18 and 6.19) generalize to arbitrary graphs. Similarly all results which we developed for 𝑀-block cyclic graphs (e.g., Theorems 6.20 and 6.21) generalize to graphs that are subject only to the eigenvalue constraint (6.56) and not the eigenvector constraint (6.57).

Section 6.17 concludes the chapter. Sections 6.18.1 - 6.18.4 provide supplementary theorems and detailed proofs of some theorems presented in this chapter

The content of this chapter is mainly drawn from [172, 173], and parts of it have been presented in [174, 175, 171, 177].

6.1.2 Notation

Given a graph,A represents the adjacency matrix of the graph. We often refer to a graph with adjacency matrix A as “graphA” for convenience. Throughout the chapter,𝑁 denotes the size of the graph and length of the signal and𝑀 denotes the decimation ratio or the number of filters in a graph filter bank, according to context.

Hence, A∈C^𝑁×𝑁. In this chapter, the (𝑖, 𝑗)^{𝑡 ℎ} block of the adjacency matrix Ais denoted by (A)𝑖, 𝑗 and (v)𝑖 denotes the (𝑖)^{𝑡 ℎ} block of the vectorv. Throughout the chapter, when it is not indicated, it should be clear that (A)𝑖, 𝑗 ∈C^{(𝑁/𝑀)×(}^𝑁/𝑀) and (v)𝑖 ∈C^𝑁/𝑀. Otherwise, they are clearly indicated to have the specified sizes. The

cyclic permutation matrix of size𝑁 is denoted byC𝑁, and it is defined as:

C𝑁 =







0 0 · · · 0 1 1 0 · · · 0 0 0

... ...

.. .

.. . ..

. ..

. 0 0 0 · · · 0 1 0







∈C^𝑁×𝑁. (6.1)

6.1.3 Review of DSP on Graphs

In DSP on graphs, the adjacency matrix of the graph of interest is considered to be the unit shiftoperator for a signal on the graph [152]. Namely, letxbe a signal on a graph with the adjacency matrixA. Then the signalycomputed as

y=A x, (6.2)

is called as the unit shifted version of x. We also would like to indicate that the adjacency matrix is not the only choice for the shift operator in general. The study in [68] proposes alternative definitions for the shift operator. Nevertheless, for simplicity, we will stick with the adjacency matrix as done in [152, 153].

In general, any square matrix of size𝑁,H∈C^𝑁×𝑁, is considered as alinear graph filteron the graph. When we havey=H x, we callyas the filtered version of the signalx. In this study, we are interested in a special type of linear filters, namely polynomial filters, which are defined as follows.

Definition 6.1(Polynomial filters [152, 126]). A linear systemH on a graphAis said to be apolynomialfilter if

H=𝐻(A) =

𝐿

𝑘=0

ℎ_𝑘 A^𝑘, (6.3)

for a set of ℎ_𝑘 ∈C. Here 𝐿is called theorderof the filter.

We can assume without loss of generality that 𝐿 < 𝑁. This is because, according to Cayley-Hamiltion theorem, powers 𝐴^𝑘 for 𝑘 ≥ 𝑁 can be expressed as linear combinations of smaller powers [85].1

For a graph with the adjacency matrix A, let the following denote the Jordan decomposition [85, 152] of the adjacency matrix

A=V J V^-1, (6.4)

1Also see Theorem 3 of [152].

whereVis composed of the (generalized) eigenvectors of the adjacency matrix and Jis the Jordan normal form ofA. When Ais diagonalizable, (6.4) reduces to the following form

A=V𝚲V^-1, (6.5)

for some diagonal𝚲consisting of the eigenvalues and some invertible square matrix V consisting of the eigenvectors of the adjacency matrix. When A has distinct eigenvalues, it is necessarily diagonalizable, but not vice versa.

Using the Jordan decomposition in (6.4), we then have the following definitions.

Definition 6.2(Graph Fourier transform [152, 153]). Letxbe a signal on a graph with the adjacency matrixA. Then the graph Fourier transform ofxon the graphA is given by

bx=V^-¹x, (6.6)

whereVhas the (generalized) eigenvectors ofAas in(6.4).

Definition 6.3(Frequency domain operation). LetHbe a linear filter on a graph with the adjacency matrix A. Then thefrequency domain operatorcorresponding toHis defined by

Hb =V^-¹H V, (6.7)

whereVhas the (generalized) eigenvectors ofAas in(6.4).

Definiton 6.3 does not imply that V diagonalizes the filter H, that is, Hb is not diagonal in general.

Notice that Definitions 6.2 and 6.3 are consistent with each other, that is, for a graph signalxand a linear filterH, we havey=Hxif and only ifby=Hbbx. As explained in Section 6.8 (and Definition 6.7) later,Hbwill be referred to as thefrequency response ofHonly whenHb is a diagonal matrix.

6.2 Building Blocks for Multirate Processing on Graphs

Dalam dokumen Signals on Networks: Random Asynchronous and Multirate Processing, and Uncertainty Principles (Halaman 183-188)