ISyE8843A, Brani Vidakovic Handout 20 1 Basics of Wavelets The first theoretical results in wavelets are connected with continuous wavelet decompositions of L2 func- tions and go back to the early 1980s. Papers of Morlet et al. (1982) and Grossmann and Morlet (1985) were among the first on this subject. Let Ãa;b(x); a 2 Rnf0g; b 2 R be a family of functions defined as translations and re-scales of a single function Ã(x) 2 L2(R); µ ¶ 1 x ¡ b Ãa;b(x) = p à : (1) jaj a 1 Normalization by p ensures that jjÃa;b(x)jj is independent of a and b: The function à (called the jaj wavelet function or the mother wavelet) is assumed to satisfy the admissibility condition, Z jª(!)j2 Cà = d! < 1; (2) R j!j R ¡ix! where ª(!) = R Ã(x)e dx is the Fourier transformation of Ã(x): The admissibility condition (2) implies Z 0 = ª(0) = Ã(x)dx: R R ® Also, if Ã(x)dx = 0 and (1 + jxj )jÃ(x)jdx < 1 for some ® > 0; then Cà < 1: Wavelet functions are usually normalized to “have unit energy”, i.e., jjÃa;b(x)jj = 1: For any L2 function f(x), the continuous wavelet transformation is defined as a function of two variables Z CWT f (a; b) = hf; Ãa;bi = f(x)Ãa;b(x)dx: Here the dilation and translation parameters, a and b, respectively, vary continuously over Rnf0g £ R: Resolution of Identity. When the admissibility condition is satisfied, i.e., Cà < 1; it is possible to find the inverse continuous transformation via the relation known as resolution of identity or Calderon’s´ reproducing identity, Z 1 da db f(x) = CWT f (a; b)Ãa;b(x) 2 : Cà R2 a If a is restricted to R+; which is natural since a can be interpreted as a reciprocal of frequency, (2) becomes Z 1 jª(!)j2 Cà = d! < 1; (3) 0 ! 1 and the resolution of identity relation takes the form Z Z 1 1 1 1 f(x) = CWT f (a; b)Ãa;b(x) 2 da db: (4) Cà ¡1 0 a Next, we list a few important properties of continuous wavelet transformations. Shifting Property. If f(x) has a continuous wavelet transformation CWT f (a; b); then g(x) = f(x¡¯) has the continuous wavelet transformation CWT g(a; b) = CWT f (a; b¡ ¯): Scaling Property. If f(x) has a¡ continuous¢ wavelet transformation ¡ ¢ p1 x a b CWT f (a; b); then g(x) = s f s has the continuous wavelet transformation CWT g(a; b) = CWT f s ; s : Both the shifting property and the scaling property are simple consequences of changing variables under the integral sign. Energy Conservation. From (4), Z 1 Z 1 Z 1 2 1 2 1 jf(x)j dx = jCWT f (a; b)j 2 da db: ¡1 Cà ¡1 0 a p1 x0¡b Localization. Let f(x) = ±(x ¡ x0) be the Dirac pulse at the point x0: Then, CWT f (a; b) = a Ã( a ): Reproducing Kernel Property. Define K(u; v; a; b) = hÃu;v;Ãa;bi: Then, if F (u; v) is a continuous wavelet transformation of f(x), Z Z 1 1 1 1 F (u; v) = K(u; v; a; b)F (a; b) 2 da db; Cà ¡1 0 a i.e., K is a reproducing kernel. The associated reproducing kernel Hilbert space (RKHS)R isR defined as a 2 1 1 1 2 da db CWT image of L2(R) – the space of all complex-valued functions F on R for which jF (a; b)j 2 Cà ¡1 0 a is finite. R Characterization of Regularity. Let (1 + jxj) jÃ(x)j dx < 1 and let ª(0) = 0: If f 2 C®(Holder¨ space with exponent ®), then ®+1=2 jCWT f (a; b)j · Cjaj : (5) Conversely, if a continuous and bounded function f satisfies (5), then f 2 C®: Example 1.1 Mexican hat or Marr’s wavelet. The function 2 d 2 2 Ã(x) = [¡e¡x =2] = (1 ¡ x2)e¡x =2 dx2 is a wavelet [known as the “Mexican hat” or Marr’s wavelet. By direct calculation one may obtain Cà = 2¼: 2 a 6 r r r r r r r r r r r r r r r r rrr r r r r r r rrrrrrrrrrrrrrrrrrrrrrrrrr r r r r r r r r r r r r rrrrrrrrrrrrr r r r r r r rrrrrrr rrrr r r r rrr r - 0 b Figure 1: Critical Sampling in R £ R+ half-plane (a = 2¡j and b = k 2¡j). d 1 1 Example 1.2 Poisson wavelet. The function Ã(x) = ¡(1+ dx ) ¼ 1+x2 is a wavelet [known as the Poisson wavelet. The analysis of functions with respect to this wavelet is related to the boundary value problem of the Laplace operator. The continuous wavelet transformation of a function of one variable is a function of two variables. Clearly, the transformation is redundant. To “minimize” the transformation one can select discrete values of a and b and still have a transformation that is invertible. However, sampling that preserves all information about the decomposed function cannot be coarser than the critical sampling. The critical sampling (Fig. 1) defined by a = 2¡j; b = k2¡j; j; k 2 Z; (6) will produce the minimal basis. Any coarser sampling will not give a unique inverse transformation; that is, the original function will not be uniquely recoverable. Moreover under mild conditions on the wavelet j=2 j function Ã, such sampling produces an orthogonal basis fÃjk(x) = 2 Ã(2 x ¡ k); j; k 2 Zg: There are other discretization choices. For example, selecting a = 2¡j; b = k will lead to non- decimated (or stationary) wavelets. For more general sampling, given by ¡j ¡j a = a0 ; b = k b0 a0 ; j; k 2 Z; a0 > 1; b0 > 0; (7) numerically stable reconstructions are possible if the system fÃjk; j; k 2 Zg constitutes a frame. Here à ! ¡j j=2 x ¡ k b0 a0 j=2 j Ãjk(x) = a0 à ¡j = a0 Ã(a0x ¡ k b0); a0 is (1) evaluated at (7). Next, we consider wavelet transformations (wavelet series expansions) for values of a and b given by (6). An elegant theoretical framework for critically sampled wavelet transformation is Mallat’s Multiresolution Analysis (Mallat, 87; 89a, 89b, 98). 3 1.1 Multiresolution Analysis A multiresolution analysis (MRA) is a sequence of closed subspaces Vn; n 2 Z in L2(R) such that they lie in a containment hierarchy ¢ ¢ ¢ ½ V¡2 ½ V¡1 ½ V0 ½ V1 ½ V2 ½ ¢ ¢ ¢ : (8) The nested spaces have an intersection that contains the zero function only and a union that is dense in L(R); \nVj = f0g; [jVj = L2(R): [With A we denoted the closure of a set A]. The hierarchy (8) is constructed such that (i) V -spaces are self-similar, j f(2 x) 2 Vj iff f(x) 2 V0: (9) and (ii) there exists a scaling function Á 2 V0 whose integer-translates span the space V0; ( ) X V0 = f 2 L2(R)j f(x) = ckÁ(x ¡ k) ; k 1 and for which the set fÁ(² ¡ k); k 2 Zg is an orthonormal basis. R Mild technical conditions on Á are necessary for future developments. It can be assumed that Á(x)dx ¸ 0: Later, we will prove that this integral is in fact equal to 1. Since V0 ½ V1, the function Á(x) 2 V0 can be represented as a linear combination of functions from V1; i.e., X p Á(x) = hk 2Á(2x ¡ k); (10) k2Z for some coefficients hk; k 2 Z: This equation is called the scaling equation (or two-scale equation) and it is fundamental in constructing, exploring, and utilizing wavelets. ADD BASES FOR Vj. ANY L2 can be projected on Vj In the wavelet literature, the reader may encounter an indexing of the multiresolution subspaces, which is the reverse of that in (8), ¢ ¢ ¢ ½ V2 ½ V1 ½ V0 ½ V¡1 ½ V¡2 ½ ¢ ¢ ¢ : (11) FORM OF Ájk(x): 1It is possible to relax the orthogonality requirement. It is sufficient to assume that the system of functions fÁ(² ¡ k); k 2 Zg constitutes a Riesz basis for V0: 4 Theorem 1.1 For the scaling function it holds Z Á(x)dx = 1; R or, equivalently, ©(0) = 1; R ¡i!x where ©(!) is Fourier transformation of Á, R Á(x)e dx: The coefficients hn in (10) are important in connecting the MRA to the theory of signal processing. The (possibly infinite) vector h = fhn; n 2 Zg will be called a wavelet filter. It is a low-pass (averaging) filter as will become clear later by considerations in the Fourier domain. To further explore properties of multiresolution analysis subspaces and their bases, we will often work in the Fourier domain. Define the function m0 as follows: X 1 ¡ik! 1 m0(!) = p hke = p H(!): (12) 2 2 k2Z The function in (12) is sometimes called the transfer function and it describes the behavior of the associated filter h in the Fourier domain. Notice that the function m0 is periodicp with the period 2¼ and that the filter taps fhn; n 2 Zg are the Fourier coefficients of the function H(!) = 2 m0(!): In the Fourier domain, the relation (10) becomes ³! ´ ³! ´ ©(!) = m © ; (13) 0 2 2 where ©(!) is the Fourier transformation of Á(x): Indeed, Z 1 ©(!) = Á(x)e¡i!xdx ¡1 X p Z 1 ¡i!x = 2 hk Á(2x ¡ k)e dx ¡1 k Z X h 1 = pk e¡ik!=2 Á(2x ¡ k)e¡i(2x¡k)!=2d(2x ¡ k) 2 k ¡1 X h ³! ´ = pk e¡ik!=2 © 2 2 k ³! ´ ³! ´ = m © : 0 2 2 By iterating (13), one gets Y1 ³ ! ´ ©(!) = m ; (14) 0 2n n=1 which is convergent under very mild conditions on rates of decay of the scaling function Á: There are several sufficient conditions for convergence of the product in (14).