Wavelet Analysis

The task of identifying the properties of an acquired signal starts with its given representation as an element in the reference (i.e. fiducial, central) representation space $\mathbf{V}_0$ . One singles out the large overall features by projecting it onto the next subspace $\mathbf{V}_1$ . This projection process suppresses the finer details of the signal. They are no longer present when the signal is represented as an element of $\mathbf{V}_1$ . Using the pyramid algorithm one repeats this process iteratively. In this process one moves from the resolution $2^{-k}$ of $\mathbf{V}_k$ to the lower resolution $2^{-(k+1)}$ of $\mathbf{V}_{k+1}\subset \mathbf{V}_k.$

To keep track of the finer details suppressed by this process, one introduces $\mathbf{O}_{k+1}$ , the orthogonal complement of $\mathbf{V}_{k+1}$ in $\mathbf{V}_k$ :

$$\mathbf{V}_{k+1}\bot \mathbf{O}_{k+1}~.$$

Thus any signal $f$ represented in $\mathbf{V}_k$ is the unique sum of the signal represented in $\mathbf{V}_{k+1}$ and its suppressed detail which lies in $\mathbf{O}_{k+1}$ :

$$P_{\mathbf{V}_k}f=P_{\mathbf{V}_{k+1}}f+P_{\mathbf{O}_{k+1}}f~.$$

In brief,

$$\mathbf{V}_k=\mathbf{V}_{k+1}\oplus \mathbf{O}_{k+1}~.$$

The o.n. bases for $\mathbf{V}_k$ and $\mathbf{V}_{k+1}$ , and hence the representations of the signal $f$ in these spaces, are known and expressed in terms of the scaling function $\phi(t)$ . These bases determine a unique basis for $\mathbf{O}_k$ whose purpose is to keep track of the suppressed $P_{\mathbf{O}_{k+1}}f$ of the signal $f$ . The process of constructing the $\mathbf{O}_{k+1}$ -basis resembles that for $\mathbf{V}_k$ and $\mathbf{V}_{k+1}$ . One starts with a square-integrable function $\psi(t)$ , the ``mother wavelet''. By applying translations and dilations to it, one obtains the desired o.n. basis for $\mathbf{O}_{k+1}$ , the space of details at resolution $2^{-(k+1)}$ . The crucial part of this endeavor is the construction of the mother wavelet from the scaling function of the MSA. The construction is done by means of the following theorem by Mallat:

Theorem 26.1 (Wavelet generation theorem)

1. Let
   $$\cdots\supset\mathbf{V}_k \supset\mathbf{V}_{k+1}\supset\cdots$$
   be the hierarchy of vector spaces which make up the MSA whose scaling function is $\phi(t)$ and whose corresponding pyramid algorithm is based on the filtering function
   $$H(\omega)=\frac{\sqrt{2}}{2}\sum_{\ell=-\infty}^\infty h_\ell e^{i\omega\ell}$$

2. Let $\psi(t)$ be a function whose Fourier transform is given by

   $$\hat\psi(\omega) = G\left(\frac{\omega}{2}\right)\hat\phi\left(\frac{\omega}{2}\right)$$
   where

   $$G(\omega) =e^{i\omega}\overline{H(\omega+\pi)}~,$$

   
   

then

I.

$$\{ \sqrt{2^{-k}} \psi(2^{-k} t-\ell):~\ell=0,\pm 1,\cdots\}$$ (2115)

is an o.n. basis for $\mathbf{O}_k$ and

II.

$$\{\sqrt{2^{-k}}\psi(2^{-k} t-\ell):~\ell,k=0,\pm 1,\cdots\}$$ (2116)

is an o.n. basis for $L^2(-\infty ,\infty )$ .

The validation of this theorem is a three step process.

1. First of all notice that the set of functions, Eq.(2.115), being orthogonal,

   $$\delta_{\ell\ell'} =\int\limits_{-\infty}^\infty \sqrt{2^{-k}} \overline\psi(2^{-k} u-\ell)\sqrt{2^{-k}} \psi(2^{-k} u-\ell') du$$

   $$=\int\limits_{-\infty}^\infty \overline\psi(u-\ell)\psi(u-\ell')du~,$$

   
   
   is equivalent to the statement that

   $$\sum_{n=-\infty}^\infty\left\vert \hat\psi(\omega+2\pi n\right\vert^2=\frac{1}{2\pi} \quad -\infty<\omega<\infty~.$$ (2117)

   
   
   The reasoning is identical to that leading to Eq.(2.112).
2. Secondly note that $\mathbf V_{k+1}\subset\mathbf V_k$ , and hence

   $$\mathbf V_k =\mathbf V_{k+1}\oplus \mathbf O_{k+1}\quad \textrm{where} ~\mathbf V_{k+1}\perp \mathbf O_{k+1}~,$$
   with

   $$\mathbf V_{k+1} =span\{\sqrt{2^{-k}} \phi(2^{-k} t-\ell) \}$$

   $$\mathbf O_{k+1} =span\{\sqrt{2^{-k}} \psi(2^{-k} t-\ell) \}~,$$

   
   
   implies that any basis element of $\mathbf V_{k+1}$ or of $\mathbf O_{k+1}$ is a linear combination of the basis elements of $\mathbf V_{k}$ . Applying this fact to the case $k=-1$ , one has

   $$\phi\left(\frac{t}{2}\right) = \sqrt{2}\sum_{-\infty}^\infty h_\ell\phi(t-\ell)$$

   $$\psi\left(\frac{t}{2}\right) = \sqrt{2}\sum_{-\infty}^\infty g_\ell\phi(t-\ell)~.$$

   
   
   The corresponding Fourier transformed equations are

   $$\hat\phi(2\omega) =H(\omega)\hat\phi(\omega)\quad \textrm{with}~ H(\omega) =\frac{\sqrt{2}}{2}\sum_{-\infty}^\infty h_\ell e^{i\omega\ell}$$ (2118)

   $$\hat\psi(2\omega) =G(\omega)\hat\psi(\omega)\quad \textrm{with}~ G(\omega) =\frac{\sqrt{2}}{2}\sum_{-\infty}^\infty g_\ell e^{i\omega\ell}$$ (2119)

   
   

3. Thirdly note that the orthogonality condition, Eq.(2.117), when combined with Eq.(2.119), yields a normalization condition on $G(\omega)$ analogous to Eq.(2.114) on page ,
   $$\left\vert G(\omega)\right\vert^2 +\left\vert G(\omega+\pi)\right\vert^2=1~.$$
   This is not the only constraint that $G$ must satisfy. One must also take into account that $\mathbf{O}_{k+1}$ is the orthogonal complement of $\mathbf{V}_{k+1}$ in $\mathbf{V}_{k}$ This fact, which is expressed by
   $$\int\limits_{-\infty}^\infty \overline\phi(u-\ell)\psi(u-\ell')du=0 \quad\textrm{for all integers }\ell\textrm{ and }\ell'~,$$
   is equivalent to

   $$\sum_{n=-\infty}^\infty \hat \phi(\omega +2\pi n)\hat \psi(\omega +2\pi n)=0 \quad -\infty<\omega<\infty$$ (2120)

   
   
   Inserting Eqs.(2.118) and (2.119) into Eq.(2.120), using the fact that $H$ and $G$ are $2\pi$ -periodic,

   $$H(\omega+2\pi n) =H(\omega)$$

   $$G(\omega+2\pi n) =G(\omega)$$

   
   
   and taking advantage of Eq.(2.112), one finds that the additional constraint on $G$ is
   $$\overline H \left(\frac{\omega}{2}\right)G \left(\frac{\omega}{2}... ...ne H \left(\frac{\omega}{2}+ \pi\right)G \left(\frac{\omega}{2}+\pi\right)=0~.$$
   Thus the filter functions $H$ and $G$ satisfy

   $$\overline H(\omega) G(\omega) +\overline H(\omega+\pi) G(\omega+\pi) =0$$ (2121)
   and

   $$\overline G(\omega) G(\omega) +\overline G(\omega+\pi) G(\omega+\pi) =1$$ (2122)

   $$\overline H(\omega) H(\omega) +\overline H(\omega+\pi) H(\omega+\pi) =1~.$$ (2123)

   
   
   A good way of remembering these constraints is that the matrix
   $$\left[ \begin{array}{cc} H(\omega)&G(\omega)\\ H(\omega+\pi)&G(\omega+\pi) \end{array}\right]$$
   is unitary. These constraints are useful if for no other reasons than that they (i) place the two sequences of Fourier coefficients $\{ h_\ell\}$ and $\{ g_\ell\}$ on certain quadratic surfaces in the Hilbert space $\ell ^2$ and that they (ii) establish a tight relation between the $h_\ell$ 's and the $g_\ell$ 's. Indeed, from Eq.(2.121) one finds
   $$G(\omega)=\overline H(\omega+\pi)e^{-i\omega}~.$$
   This relation is not unique. Other possibilities are
   $$G(\omega)=\overline H(\omega+\pi)e^{-(2p+1)i\omega}\quad\textrm{where }p\textrm{ is an arbitrary integer}~.$$
   Each side of this equation is a Fourier series in powers of $e^{i\omega}$ . Equating equal powers one finds
   $$g_\ell=(-1)^\ell~ h_{2p+1-\ell}~.$$
   With both $G(\omega)$ and $H(\omega)$ at hand, one solves the two Eqs.(2.118) and (2.119). The solutions are

   $$\hat\phi(\omega )=\hat\phi(0)\prod\limits_{k=1}^\infty H\left(\frac{\omega}{2^k}\right)$$

   $$\hat\psi(\omega )=\hat\phi(0)G(\omega)\prod\limits_{k=1}^\infty H\left(\frac{\omega}{2^k}\right)~.$$

   
   
   The inverse Fourier transform of these solutions yields the sought after scaling function

   $$\phi(t) =\hat\phi(0)\int\limits_{-\infty}^\infty \prod\limits_{k=1}^\infty H\left(\frac{\omega}{2^k}\right) \frac{e^{i\omega t}}{\sqrt{2\pi}}d\omega$$
   and the mother wavelet

   $$\psi(t) =\hat\phi(0)\int\limits_{-\infty}^\infty G(\omega) \prod\limits_{... ...fty H\left(\frac{\omega}{2^k}\right) \frac{e^{i\omega t}}{\sqrt{2\pi}}d\omega~.$$

   
   
   Shifting and dilating this mother wavelet yields the o.n. basis functions, Eq.(2.115), for $\mathbf{O}_k$

Exercise 26.8 (ORTHOGONALITY OF THE DETAIL SPACES)
Validate conclusion # II. of the theorem on page , i.e. point out why, whenever $k\ne k'$ , the functions in the space of details $\mathbf{O}_k$ are orthogonal to the functions in the space of details $\mathbf{O}_{k'}$ .

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