CHAPTER THREE Basic Orthogonal Wavelet Theory In Chapter 2 we saw how multiresolution analysis (MRA) works for the Haar sys- tem. A signal was decomposed into many components on different resolution levels. These components are mutually orthogonal. Despite their attractiveness, the Haar scalets and wavelets are not continuous functions. The discontinuities can create problems when applied to physical modeling. In this chapter we will construct many other orthogonal wavelets that are continuous and may even be smooth functions. Yet they preserve the same MRA and orthogonality as the Haar wavelets do. The wavelet basis consists of scalets ϕ m,n (t) = 2 m/2 ϕ(2 m t − n), m, n ∈ Z , and wavelets ψ m,n (t) = 2 m/2 ψ(2 m t − n), m, n ∈ Z . 3.1 MULTIRESOLUTION ANALYSIS The study of orthogonal wavelets begins with the MRA. In this section we will show how an orthonormal basis of wavelets can be constructed starting from a such mul- tiresolution analysis. Assume that a scalet ϕ is r times differentiable with rapid decay ϕ (k) (t) ≤ C pk (1 +|t |) −p , k = 0, 1, 2, ,r, (3.1.1) p ∈ Z, t ∈ R, C pk − constants. Thus we have defined a set, S r , which will be used in the text; ϕ ∈ S r ={ϕ : ϕ (k) (t) exist with rapid decay as in (3.1.1)}. 30 Wavelets in Electromagnetics and Device Modeling. George W. Pan Copyright ¶ 2003 John Wiley & Sons, Inc. ISBN: 0-471-41901-X MULTIRESOLUTION ANALYSIS 31 A multiresolution analysis of L 2 (R) is defined as a nested sequence of closed sub- spaces {V j } j∈Z of L 2 (R), with the following properties [1]: (1) ···⊂V −1 ⊂ V 0 ⊂···⊂L 2 (R). (2) f (·) ∈ V m ↔ f (2·) ∈ V m+1 . (3) f (t) ∈ V 0 ⇒ f (t + n) ∈ V 0 for all n ∈ Z. (4)  m V m = 0, closure(  m V m ) = L 2 (R). (5) There exists ϕ(t) ∈ V 0 such that set {ϕ(t − n)} forms a Riesz basis of V 0 . A Riesz basis of a separable Hilbert space H is a basis { f n } that is close to being orthogonal. That is, there exists a bounded invertible operator which maps { f n } onto an orthonormal basis. Let us explain these mathematical properties intuitively: • In property (1) we form a nested sequence of closed subspaces. This sequence represents a causality relationship such that information at a given level is suf- ficient to compute the contents of the next coarser level. • Property (2) implies that V j is a dilation invariant subspace. As will be seen in later sections, this property allows us to build multigrid basis functions accord- ing to the nature of the solution. In the rapidly varying regions the resolution will be very fine, while in the slowly fluctuating regions the bases will be coarse. • Property (3) suggests that V j is invariant under translation (i.e., shifting). • Property (4) relates residues or errors to the uniform Lipschitz regularity of the function, f , to be approximated by expansion in the wavelet bases. • In property (5) the Riesz basis condition will be used to derive and prove conver- gence. The last two properties are more suitable for mathematicians; interested readers are referred to [1–3]. Clearly, √ 2ϕ(2t −n) is an orthonormal basis for V 1 , since the map f  √ 2 f (2·) is isometric from V 0 onto V 1 . Since ϕ ∈ V 1 ,wehave ϕ(t) =  k h k √ 2ϕ(2t − k), {h k }∈l 2 , t ∈ R. (3.1.2) Equation (3.1.2) is called the dilation equation, and is one of the most useful equa- tions in the field of wavelets. The MRA allows us to expand a function f (t) in terms of basis functions, consisting of the scalets and wavelets. Any function f ∈ L 2 (R) can be projected onto V m by means of a projection operator P V m ,defined as P V m f = f m :=  n f m,n ϕ m,n ,where f m,n is the coefficient of expansion of f on the basis ϕ m,n . From the previously listed MRA properties, it can be proved that lim m→∞ || f − f m || = 0, that is to say, that by increasing the resolution in MRA, a function can be approximated with any precision. 32 BASIC ORTHOGONAL WAVELET THEORY 3.2 CONSTRUCTION OF SCALETS ϕ(τ) Haar wavelets are the simplest wavelet system, but their discontinuities hinder their effectiveness. Naturally people have found it useful to switch from a piecewise con- stant “box” to a piecewise linear “triangle.” Unfortunately, the triangles are no longer orthogonal. Thus an orthogonalization procedure must be conducted, which leads to the Franklin wavelets. 3.2.1 Franklin Scalet Consider a triangle function depicted in Fig. 3.1. θ(t) = (1 −|t − 1 |)χ [0,2] (t). This function is the convolution of two pulse functions of χ [0,1] (t),whereχ [0,1] (t) is the characteristic function that is 1 in [0, 1] and 0 outside this interval. The Fourier transform of the pulse function can be obtained using the following relationships: {1(t) − 1(t − 1)}↔ 1 s (1 − e −s ) = 1 iω (1 − e −iω ), where 1(t) is the Heaviside step function. By the convolution theorem, the triangle has as its Fourier transform  1 − e −iω iω  2 = e −iω  e iω/2 − e −iω/2 iω  2 = e −iω  sin ω/2 ω/2  2 = ˆ θ(ω). Notice that θ(t) is centered at t = 1. Let us define θ c (t) := θ(t + 1), a triangle centered at t = 0 with a real spectrum of ˆ θ c (ω) =  sin ω/2 ω/2  2 . Occasionally we will use T (t) := θ c (t) to denote the triangle centered at the origin. To find the orthogonal function ϕ(t), we employ the isometric property of the Fourier transform. First, we may show that  ∞ −∞ ϕ(t − n)ϕ(t) dt = 1 2π  ∞ −∞ dω ˆϕ(ω) ˆϕ(ω)e iωn , (3.2.1) where the overbar denotes the complex conjugate. CONSTRUCTION OF SCALETS ϕ(τ ) 33 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t θ (t) FIGURE 3.1 The triangle function θ(t). Show. LHS =  ∞ −∞  1 2π  e it e −in ˆϕ()d  1 2π  e iωt ˆϕ(ω)dω  dt = 1 2π  dωd dt ˆϕ(ω)e −in ˆϕ() 1 2π e i(ω+)t = 1 2π  dω ˆϕ(ω)  de −in ˆϕ() 1 2π  dte i(ω+)t = 1 2π  dω ˆϕ(ω)  de −in ˆϕ()δ(ω +) = 1 2π  dω ˆϕ(ω) ˆϕ(−ω)e inω , where δ(·) is the Dirac delta. Since ϕ(t) is real, ˆϕ(ω) = ˆϕ(−ω), that is, ˆϕ(−ω) = ˆϕ(ω). Hence LHS = 1 2π  dω ˆϕ(ω) ˆϕ(ω)e inω . The orthogonality may be derived from the time domain by considering two basis functions. If ϕ(t) and ϕ(t − n) make an orthonormal system, then δ 0,n =  ∞ −∞ ϕ(t − n)ϕ(t) dt, 34 BASIC ORTHOGONAL WAVELET THEORY where δ 0,n is the Kronecker delta. By employing (3.2.1), we arrive at δ 0,n = 1 2π  ∞ −∞ dωe iωn ˆϕ(ω) ˆϕ(ω) = 1 2π ∞  k=−∞  2π 0 |ˆϕ(ω + 2kπ) | 2 e iωn dω = 1 2π  2π 0  k |ˆϕ(ω +2kπ)| 2 e iωn dω. (3.2.2) We define a periodic function f (ω) =|ˆϕ † (ω) | 2 :=  k |ˆϕ(ω + 2kπ) | 2 . The Fourier series of a periodic function with period of 2π is f (ω) = c 0 + ±∞  n=1 c n e iωn . (3.2.3) Comparing (3.2.2) with (3.2.3), we conclude that c 0 = 1, and c n = 0forn = 1. This conclusion can also be drawn from the uniqueness of the Fourier transform as follows. We know that 1 2π  2π 0 e iωn dω =  1ifn = 0 0ifn = 0; or equivalently 1 2π  2π 0 e iωn dω = δ 0,n . On the other hand, (3.2.2) suggests that [  2π 0  k |ˆϕ(ω+2kπ) | 2 e iωn dω]/2π = δ 0,n . From the uniqueness of the Fourier transform, we conclude that |ˆϕ † (ω) | 2 :=  k |ˆϕ(ω +2kπ)| 2 = 1. (3.2.4) In the following paragraphs we will construct the scalet ϕ(t) using translated trian- gles θ(t + 1 −n) as building blocks. Since ϕ ∈ V 0 ,wehaveϕ(t) =  n a n θ(t + 1 − n) for a sequence {a n }∈l 2 , meaning that  n |a n | 2 < +∞. Taking the Fourier transform, we immediately have ˆϕ(ω) =  n a n e iω(1−n) ˆ θ(ω) =  n a n e −iωn ˆ θ c (ω) (3.2.5) = α(ω) ˆ θ c (ω), CONSTRUCTION OF SCALETS ϕ(τ ) 35 where α(ω) =  n a n e −iωn . Hence |ˆϕ † (ω) | 2 =|α(ω) | 2 | ˆ θ † c (ω) | 2 = 1, (3.2.6) where | ˆ θ † c (ω) | 2 =  k | ˆ θ c (ω + 2kπ)| 2 . Equation (3.2.6) can be used to find α(ω).Bydefinition, we have |ˆϕ † (ω) | 2 =  k |ˆϕ(ω + 2kπ)| 2 =  k |α(ω + 2kπ) ˆ θ c (ω + 2kπ)| 2 . Since α(ω) =  n a n e −inω , we have α(ω + 2kπ) =  n a n e in(ω+2kπ) =  n a n e inω = α(ω). Thus |ˆϕ † (ω) | 2 =     α(ω)     2  k     ˆ θ c (ω + 2kπ)     2 =|α(ω) | 2 | ˆ θ c † (ω) | 2 . Later in this section we show that | ˆ θ c † (ω) | 2 can be found in a closed form | ˆ θ c † (ω) | 2 =  k     sin(ω + 2kπ)/2 (ω + 2kπ)/2     4 . Therefore |α(ω) | 2 = 1  k     sin(ω + 2kπ)/2 (ω + 2kπ)/2     4 . 36 BASIC ORTHOGONAL WAVELET THEORY It will be seen in the next paragraph that  k     sin(ω + 2kπ)/2 (ω + 2kπ)/2     4 = 1 − 2 3 sin 2 ω 2 . (3.2.7) Hence ˆϕ(ω) = α(ω) ˆ θ c (ω) = α(ω)  sin ω/2 ω/2  2 = 1  1 − 2 3 sin 2 ω/2  sin ω/2 ω/2  2 . (3.2.8) Let us derive (3.2.7). The inverse Fourier transform of e iωk is 1 2π  π −π e iωx e iωk dω = 1 2π  π −π dωe iω(x+k) = 1 2π e iω(x+k) i(x + k)      π ω=−π = e i(x+k)π − e −i(x+k)π 2i 1 π(x + k) = sin π(x + k) π(x + k) . Parseval’s law relates the energy of a signal in the spatial domain and spectral domain as 1 2π  π −π |e iωk | 2 dω =  k     sin π(x +k) π(x + k)     2 =|sin π x | 2  k 1 |π(x + k) | 2 . Notice that the left-hand side of the previous equation is 1. So we have 1 sin 2 π x =  k 1 [π(x + k)] 2 . (3.2.9) Taking the second derivative of the previous equation with respect to x, we obtain  1 sin 2 π x   = 6π 2 1 − 2 3 sin 2 π x sin 4 π x CONSTRUCTION OF SCALETS ϕ(τ ) 37 and   k 1 [π(x + k)] 2   = 6 π 2  k 1 (x +k) 4 . Therefore  k 1 (π x + kπ) 4 = 1 − 2 3 sin 2 π x sin 4 π x . (3.2.10) Letting π x = ω/2, we obtain from this equation that  k 1 (kπ +(ω/2)) 4 = 1 − 2 3 sin 2 (ω/2) sin 4 (ω/2) (3.2.11) which is equation (3.2.7). The coefficients a n in (3.2.5) can be evaluated numerically. As given in (3.2.5), ˆϕ(ω) =  k a k e −ikω ˆ θ c (ω) =  sin 2 (ω/2) (ω/2)  2  k a k e −iωk . Using the time shift property of the Fourier transform, we obtain ϕ(t) =  k a k θ(t + 1 −k), a k = O(e −a|k | ). (3.2.12) Notice again that θ(t + 1) := θ c (t) is a triangle centered at t = 0, and its Fourier transform θ c (ω) =  sin(ω/2) (ω/2)  2 . Coefficients a k will be evaluated as follows: From the expression α(ω) = 1  1 − 2 3 sin 2 (ω/2) (3.2.13) α(ω) is a periodic function of period 2π, which has the Fourier series  n a n e −iωn = 1  1 − 2 3 sin 2 (ω/2) . 38 BASIC ORTHOGONAL WAVELET THEORY TABLE 3.1. First Ten Coefficients of a n = a − n for the Franklin Scalet a 0 1.29167548213672 a 1 −0.17466322755518 a 2 0.03521011276878 a 3 −0.00787442432698 a 4 0.00184794571482 a 5 −4.45921398374e-04 a 6 1.09576772871e-04 a 7 −2.72730550551e-05 a 8 6.85286905090e-06 a 9 −1.73457608425e-06 −5 −4 −3 − 2 −1 0 1 2 3 4 5 −1 −0.5 0 0.5 1 1.5 2 φ (t) ψ(t) −20 −15 −10 −5 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 φ(ω) ψ(ω) FIGURE 3.2 Franklin scalet ϕ and wavelet ψ. CONSTRUCTION OF SCALETS ϕ(τ ) 39 By multiplying both sides by e iωk /2π and integrating, bearing in mind that 1 2π  π −π e iω(k−n) dω = δ k,n , we obtain a k = 1 2π  π −π e iωk  1 − 2 3 sin 2 (ω/2) dω = 1 π  π 0 cos kω  1 − 2 3 sin 2 (ω/2) dω. This equation provides a numerical expression for the evaluation of a n , which can be accomplished by imposing Gaussian–Legendre quadrature. The values of a n are displayed in Table 3.1. Using these values of a n and the translated triangle functions θ(t +1−n), the Franklin scalet is constructed according to (3.2.12). From the integral expression of a k , we observe that a −k = a k .Alsoθ c (t) is symmetric. Therefore the Franklin scalet is an even function. The Franklin wavelet is symmetric about t = 1 2 , and will be studied in the next section. The Franklin scalet and wavelets are depicted in Fig. 3.2. 3.2.2 Battle–Lemarie Scalets The Franklin wavelets employ the triangle functions as building blocks in the con- struction of an orthogonal system. These triangles are continuous functions but not smooth; their derivatives are discontinuous at certain points. If we convolve the tri- angle with the box one more time, the resulting function will be smooth. The trans- lations of this smooth function can then be used as building blocks to build smooth orthogonal wavelet systems. The greater the number of convolutions conducted, the smoother the building block functions become. This smoothness is achieved at the expense of larger support widths of the resulting scalets. In general, the B-spline of degree N is obtained by convolving the “box” N times. Hence ˆ θ N (ω) = e −iκ(ω/2)  sin(ω/2) ω/2  N +1 , where κ =  0ifN = odd 1ifN = even , and as such any shift by an integer can be ignored. We use integer translations of the basis functions, therefore only the half-integer shifts matter. The corresponding α 1 (ω) for N = 1 is the Franklin in (3.2.13). For N = 2,α 2 (ω) ={ 1 15 [2cos 4 (ω/2) + 11 cos 2 (ω/2) + 2]} −1/2 . The resulting Battle–Lemarie wavelets are illustrated in Fig. 3.3. Detailed construction of higher-order Battle–Lemarie wavelets is left to readers as an exercise problem in this chapter. [...]... last property is for wavelets, but it is stated here for ease of reference These properties will be used in later sections of this chapter to construct the Daubechies and Coifman wavelets 3.6 DAUBECHIES WAVELETS In contrast with the infinitely supported Franklin or Battle–Lemarie, the Daubechies wavelets are compactly supported orthogonal systems In the construction of Daubechies wavelets, we seek a... corresponding scalet and wavelet are illustrated in Fig 3.4 It is expected that higher order wavelets are smoother, but their supports are wider Higher-order Daubechies wavelets can be derived in the same way presented here The coefficients are tabulated in Table 3.3 and will be employed to construct the Daubechies scalets and wavelets of different orders In general, for Daubechies scalets h n = 0 for n < 0 and... Preliminary Properties of Scalets In the previous discussions we used the triangle functions as building blocks to generate the Franklin wavelets, according to ϕ(t) = k ak θc (t − k) If the triangles are replaced by smoother building blocks, higher-order Battle–Lemarie wavelets may be obtained in the same manner Unfortunately, the number of nonzero coefficients ak are infinite, although ak decays very rapidly,... For n = 15, | an | < 2 · 10−9 , so it may be truncated with 16 terms The numerical data for h n and bn are tabulated in Table 3.2, while the resulting Franklin wavelets are depicted in Fig 3.2 By the same token, the resulting Battle–Lemarie wavelets are constructed and plotted in Fig 3.3 3.5 PROPERTIES OF SCALETS ϕ(ω) ˆ The scalets ϕ(t − n) are orthonormal Furthermore ϕ(ω) is bounded, and ϕ(ω) is ˆ... support of the Coiflets of order L = 2K is 3L − 1 Consider the case L = 2 Notice that (3.7.1) states the vanishing moments for the wavelets, and (3.7.3) is the normalization of the scalet, with respect to the d.c component Both of these two equations are shared by other wavelets The unique property of the Coiflets is contained in (3.7.2), namely the vanishing moments of the scalets This property can... the filter bank coefficients: (i) k √ h k / 2 = 1 (ii) k (−1)k √ hk 2 (iii) k h k h k−2n = δ0,n = 0 The two compactly support wavelets, Daubechies and Coifman, have similarities and distinctions in the governing equations Table 3.4 summarizes and compares the nature of these wavelets Equations (i), (ii), (iii), (3.6.4), and (3.7.7) are sufficient to solve the h k for the Coiflets of order 2 They form... Therefore ˆ h(π) = 0 In general, ˆ h((2m + 1)π) = 0 and ˆ h(2mπ) = 1 ˆ Furthermore h(ω/2) is a periodic function with period 4π 3.3 WAVELET ψ(τ) After the scalets are obtained, we can create the corresponding wavelets ψ(t) In this process we may take advantage of the MRA structure by choosing {ψ(t − n)} as an orthonormal basis of W0 , which is the orthogonal complement of V0 in V1 , namely WAVELET ψ(τ ) V1... systems In the construction of Daubechies wavelets, we seek a finite set of nonzero coefficients h k in the dilation equation √ ϕ(t) = h k 2ϕ(2t − k) k Recalling that ˆ ω = h 2 k hk √ e−i(ω/2)k 2 DAUBECHIES WAVELETS 57 and the properties of ϕ(ω), ˆ ˆ h(0) = 1, ˆ h(π) = 0, we obtain (i) 1 = (ii) 0 = h √k 2 (−1)k h k √ k 2 k Furthermore δn,0 = ϕ(t)ϕ(t − n) dt √ √ h k h l 2ϕ(2t − k) 2ϕ(2t − 2n − l) dt = k... centered at √ √ ( 2/4, 2/4) with radius 1/2, and passes through the origin The last equation in (3.6.1) gives h0 1 √ − h0 + h1 2 1 √ − h1 2 = 0, that is, 1 h0 − √ 2 2 2 1 + h1 − √ 2 2 2 = 1 2 2 , DAUBECHIES WAVELETS 59 which represents the same circle as the third equation does This means that the four equations are not independent Daubechies introduced ν, such that ν(ν − 1) , D 1−ν , h1 = D 1+ν , h2 = D... (3.6.2) where D= √ 2(1 + ν 2 ), ν ∈ R (3.6.3) We can verify that for any ν, the four equations in (3.6.1) are all satisfied We need one more equation to specify ν, which will be obtained as follows The wavelets have the frequency domain expression ˆ ψ(ω) = g ˆ ω ω ϕ ˆ 2 2 We have also had the zero moment ∞ −∞ tψ(t) dt = 0 From ˆ ψ(ω) = ∞ −∞ ψ(t)e−iωt dt, we have the derivative ˆ ψ (ω) = −itψ(t)e−iωt . Haar scalets and wavelets are not continuous functions. The discontinuities can create problems when applied to physical modeling. In this chapter we will construct many other orthogonal wavelets that. preserve the same MRA and orthogonality as the Haar wavelets do. The wavelet basis consists of scalets ϕ m,n (t) = 2 m/2 ϕ(2 m t − n), m, n ∈ Z , and wavelets ψ m,n (t) = 2 m/2 ψ(2 m t − n), m, n ∈. . 3.1 MULTIRESOLUTION ANALYSIS The study of orthogonal wavelets begins with the MRA. In this section we will show how an orthonormal basis of wavelets can be constructed starting from a such mul- tiresolution