Wavelets – An Introduction

Wavelets are used in a wide range of applications such as signal analysis, signal compression, finite element methods, differential equations, and integral equations. In the following we will discuss the limitations of traditional basis expansions and show why wavelets are in many cases more efficient representations. A mathematical treatment of second generation wavelets as well as an example will be provided. 1 What are wavelets . . . and why do we care? Traditional basis expansions such as the Fourier transform and the Laplace transform have proven to be indispensable in many domains. In the last decades it has however been recognized that different limitations hamper the practicality of these representations: (L1) Localization in space The Fourier transform is localized in frequency but the global support of the basis functions prevents a localization in space1. For many applications in particular the local behaviour of signals is of interest. (L2) Faster transform algorithms In recent years the advance of data acquisition technology outpaced the available computing power significantly making the Fast Fourier Transform with its O(n log n) complexity a bottleneck in many applications. (L3) More flexibility Traditional basis expansions provide no or almost no flexibility. It is therefore usually not possible to adapt a representation to the problem at hand. An important reason for this lack of flexibility is the orthogonal nature of traditional basis expansions. (L4) Arbitrary domains Traditional basis representations can only represent functions defined of Euclidean spaces R 2. Many real-world problems have embeddings X ⊂ R as domain and it is desirable to have a representation which can be easily adapted for these spaces. (L5) Weighted measures and irregularly sampled data Traditional transforms can usually not be employed on spaces with weighted measures or when the input data is irregularly sampled. ∗[email protected] The frequency localization of the Fourier transform refers to the fact that every Fourier basis function captures characteristics of the input signal in a limited frequency band. Space localization refers to a limited effective support of the basis functions in the primary domain, for audio signals, for example, the primary or “space” domain is time. A notable exception is the sphere where, for example, Spherical Harmonics [10] provide a basis. These limitations motivated the development of wavelets. Many different fields such as applied mathematics, physics, signal processing, and computer science provided contributions and today both a thorough mathematical theory and fast and practical algorithms exist. An important distinction between traditional basis expansions and wavelets is that there is not a single set of basis functions that defines a wavelet. Instead, the members of a family of representations with vastly different properties are denoted as wavelets. Common to all of them are three properties: (P1) The sequence {fk}k=1 forms a basis or a frame3 of Lp. (P2) The elements of {fk}k=1 are localized in both space and frequency. (P3) Fast algorithms for the analysis, synthesis, and processing of signals in its basis representation exist. These three properties – and the flexibility they leave – are the key to the efficiency and versatility of wavelets. Some of the first non-trivial wavelets that have been developed are the Daubechies wavelet [4] and the Meyer wavelet [11]. These, and most other wavelets developed in the 1980s, are first generation wavelets whose construction requires the Fourier transform and whose basis functions have to be (dyadic) scales and translates of one particularmother basis function5 (cf. Section 3). The limitations L3 to L5 thus still apply for first generation wavelets. The work by Mallat and Sweldens overcame these restrictions and led to the development of second generation wavelets which will be discussed in more detail in the following section. Wavelets can be categorized into discrete (DWT) and continuous (CWT) wavelet transforms. To speak in broad terms, the basis functions of DWTs are defined over a discrete space which becomes continuous only in the limit case, whereas the basis functions of CWTs are continuous but require discretization if they are to be used on a computer; see for example the book by Antoine et al. [1] for a more detailed discussion of the differences. In signal compression applications mostly discrete wavelets are employed, whereas for signal analysis typically continuous wavelets are used. 2 Second Generation Wavelets In this section a mathematical characterization of second generation wavelets will be provided. See the paper by Sweldens [17] or the thesis by Lessig [9] for a more comprehensive treatment. Second generation wavelets permit the representation of functions in L2, the space of functions with finite energy 6, in a very general setting L2 ≡ L2(X,Σ, μ), where X ⊆ R is a spatial domain, Σ denotes a σ-algebra defined overX , and μ is a (possibly weighted) measure on Σ7. The inner product defined over X will be denoted as 〈·, ·〉. A multiresolution analysis M = {Vj ⊂ L2 | j ∈ J ⊂ Z} consisting of a sequence of nested subspaces Vj on different levels j is employed to define the basis functions. M satisfies A frame is an overcomplete representation, that is some basis functions fi can be represented as linear combination of other basis functions. See the book by Christensen [2] for more details. In the following we will only consider the space L2 of functions with finite energy. See the book by Chui [3] for a more detailed discussion. In engineering and many other disciplines, “finite energy” is often used synonymously with “square-integrable”, that is the `2 norm of all functions in the space has to be finite. For first generation wavelets,X = R and μ is the Haar-Lebesgue measure [5]. Figure 1: Father scaling basis function φ and wavelet mother basis function ψ for the Haar basis. 1. Vj ⊂ Vj+1. 2. ⋃ j∈J Vj is dense in L2. 3. For every j ∈ J , a basis of Vj is given by scaling functions {φj,k | k ∈ K(j)}. The index set K(j) is defined over all basis functions on level j. Next to the primary multiresolution analysisM, a dual multiresolution analysis M̃ = {Ṽj ⊂ L2 | j ∈ J ⊂ Z} formed by dual spaces Ṽj exists, and a basis of the Ṽj is given by dual scaling functions {φ̃j,k | k ∈ K(j)}. The primary and dual scaling functions are biorthogonal 〈φj,k, φ̃j,k′〉 = δk,k′ . The nested structure of the spaces Vj ⊂ Vj+1 implies the existence of difference spaces Wj with Vj ⊕ Wj = Vj+1. The Wj are spanned by sets of wavelet basis functions {ψj,m | m ∈M(j)}. Analogous to the spaces Ṽj , dual wavelet spaces W̃j with Ṽj⊕W̃j = Ṽj+1 exist. These are spanned by dual wavelet basis functions {ψ̃j,m | m ∈ M}. The primary and dual wavelet basis functions on all levels are biorthogonal 〈ψj,m, ψj′,m′〉 = δj,j′δk,k′ . For all levels j, the spaces Vj andWj are subspaces of Vj+1. This implies the existence of refinement relationships