5.3.1 Using Orthogonal Wavelets
One key result of the wavelet theory is that signals can be decomposed in a series of orthogonal wavelets.
This is similar to the notion of decomposing a signal in terms of discrete Fourier transform components or Walsh or Haar functions. Orthogonality insures a unique and complete representation of the signal.
Likewise the orthogonal complement provides some measure of the error in the representation. The dif- ference in terms of wavelets is that each of the orthogonal vector spaces offers component signals with varying levels of resolution and scale. This is why Mallat [24] named his algorithm the multiresolution signal decomposition. Each stage of the algorithm generates wavelets with sequentially finer representa- tions of signal content. To achieve an orthogonal wavelet representation, a given wavelet function, f(t), at a scaling index level equal to zero, is first dilated by the scale coefficient 2j, then translating it by 2−jn and normalizing by gives:
2−jφ2j(t−2−jn) (5.10b)
The algorithm begins with an operator A2jfor discrete signals that takes the projections of a signal, f(t) onto the orthonormal basis, V2j:
A2jf(t)=2−j ∞ n=−∞
f(u),φ2j(u−2−jn)φ2j(t−2−jn) (5.11)
where 2j defines the level of resolution. A2jis defined as the multi-resolution operator that approximates a signal at a resolution 2j. Signals at successively lower resolutions can be obtained by repeated application of the operator A2j(−J ≤j≤ −1), where J specifies the maximum resolution, such that A2jf(x)is the closest approximation of function f(x)at resolution 2j. Here we note that is simply a convolution defined thusly,
f(u),φ2j(u−2−jn) = ∞
−∞f(u)φ(u−2−jn)du (5.12)
Here f(x)is the impulse response of the scaling function. The Fourier transforms of these functions are lowpass filter functions with successively smaller halfband lowpass filters. This convolution synthesizes the coarse signal at a resolution/scaling level j:
C2jf = f(t),φ2j(t−2−jn) (5.13) Each level j generates new basis functions of the particular orthonormal basis with a given discrete approx- imation. In this case, larger j provides for decreasing resolution and increasing the scale in proportional fashion for each level of the orthonormal basis. Likewise, each sequentially larger j provides for time shift in accordance with scale changes, as mentioned above, and the convolution or inner product operation generates the set of coefficients for the particular basis function. A set of scaling functions at decreasing levels of resolution, j=0,−1,−2,. . .,−6 is given in Reference 25.
The next step in the algorithm is the expression of basis function of one level of resolution, f 2jby at a higher resolution, f 2j+1. In the same fashion as above, an orthogonal representation of the basis V 2jin terms of V 2j+1is possible, or
φ2j(t−2−jn)=2−j−1 ∞ k=−∞
φ2j(u−2−jn),φ2j+1(u−2−j−1k)φ2j+1(t−2−j−1k) (5.14)
Here the coefficients are once again the inner products between the two basis functions. A means of translation is possible for converting the coefficients of one basis function to the coefficients of the basis
function at a higher resolution:
C2jf = f(u),φ2j(t−2−jn)
=2−j−1 ∞ k=−∞
φ2j(u−2−jn),φ2j+1(u−2−j−1k)f(u),φ2j+1(t−2−j−1k) (5.15)
Mallat [24] also conceives of the filter function, h(n), whose impulse response provides this conversion, namely,
C2jf = f(u),φ2j(t−2−jn) =2−j−1 ∞ k=−∞
h˜(2n−k)f(u),φ2j+1(t−2−j−1k) (5.16)
where h(n)=2−j−1ãf 2j(u−2−jn), f 2j+1(u−2−j−1k)ề and ˜h(n)=h(−n)is the impulse response of the appropriate mirror filter.
Using the tools already described, Mallat [24] then proceeds to define the orthogonal complement, O2jto the vector space V2j at resolution level j. This orthogonal complement to V2jis the error in the approximation of the signal in V2j+1by use of basis function belonging to the orthogonal complement.
The basis functions of the orthogonal complement are called orthogonal wavelets, y(x), or simply wavelet functions. To analyze finer details of the signal, a wavelet function derived from the scaling function is selected. The Fourier transform of this wavelet function has the shape of a bandpass filter in frequency domain. A basic property of the function y is that it can be scaled according to
CWTx(τ, a)= 1
√a
x(at)g∗ t−τ
a
dt (5.16a)
An orthonormal basis set of wavelet functions is formed by dilating the function y(x)with a coefficient 2jand then translating it by 2−jn, and normalizing by . They are formed by the operation of convolving the scale function with the quadrature mirror filter
ψ(ω)=G ω
2 φω
2
(5.16b) where G(ω)=e−jω—H(ω+π)is the quadrature mirror filter transfer response and g(n)=(−1)1− nh(1−n)is the corresponding impulse response function.
The set of scaling and wavelet functions presented here form a duality, together resolving the temporal signal into coarse and fine details, respectively. For a given level j then, this detail signal can once again be represented as a set of inner products:
D2jf = f(t),ψ2j(x−2−jn) (5.17) For a specific signal, f(x), we can employ the projection operator as before to generate the approximation to this signal on the orthogonal complement. As before, the detail signal can be decomposed using the higher resolution basis function:
D2jf = f(u),ψ2j(t−2−jn)
=2−j−1 ∞ k=−∞
ψ2j(t−2−jn),ψ2j+1(u−2−j−1k)f(u),φ2j+1(t−2−j−1k) ´Y (5.18)
Keep alternative sample
Cd2j+1f
Dd2jf
Dd2jf Convolve
G*
Convolve H*
Keep alternative sample j=j–1 –J≤j≤–1
FIGURE 5.2 Flow chart of a multiresolution algorithm showing how successive coarse and detail components of resolution level, j, are generated from higher resolution level, j+1.
or in terms of the synthesis filter response for the orthogonal wavelet D2jf = f(u),ψ2j(t−2−jn) =2−j−1
∞ k=−∞
˜
g(2n−k)f(u),φ2j+1(t−2−j−1k) (5.19)
At this point, the necessary tools are here for a decomposition of a signal in terms of wavelet components, coarse and detail signals. Multiresolution wavelet description provides for the analysis of a signal into lowpass components at each level of resolution called coarse signals through the C operators. At the same time the detail components through the D operator provide information regarding bandpass components.
With each decreasing resolution level, different signal approximations are made to capture unique signal features. Procedural details for realizing this algorithm follow.
5.3.2 Implementation of the Multiresolution Wavelet Transform:
Analysis and Synthesis of Algorithms
A diagram of the algorithm for the multiresolution wavelet decomposition algorithm is shown in Figure 5.2. A step-by-step rendition of the analysis is as follows:
1. Start with N samples of the original signal, x(t), at resolution level j=0.
2. Convolve signal with original scaling function, f(t), to find C1 f as in Equation 5.13 with j=0.
3. Find coarse signal at successive resolution levels, j= −1,−2,. . .,−J through Equation 5.16; keep every other sample of the output.
4. Find detail signals at successive resolution levels, j= −1,−2,. . .,−J through Equation 5.19; keep every other sample of the output.
5. Decrease j and repeat steps 3 through 5 until j= −J .
Signal reconstruction details are presented in References 24 and 26.