Wavelets

Abdou Youssef

Wavelet-Based Approximation: Introduction and Motivation
Some ``Desirables'' in Approximation Theory
Illustrations of Translates and Dilates
Mathematical Formulations
Conditions for Achieving the Analysis and Synthesis Properties
Relation to Subband Coding
Illustration of Wavelets' Dynamic Adjustment to Regional Variations without Blockiness (Comparison with Whole DCT)
Mathematical Method for Computing the Four Filters
Algorithm for Computing the Taps of the Filters
Examples of Four-Filter Sets
Daubechies Orthogonal Wavelets

1. Wavelet-Based Approximation: Introduction and Motivation

2. Some ``Desirables'' in Approximation Theory

Building Block Property
Availability of a single approximating function , called a scaling function, to serve as a building block for approximations (or samplings) of any given function
- The approximation is done by having different translates of $\phi$ fitted as closely as possible to
- The translates of $\phi$ are $(\phi_k)_k$ , where $\phi_k(t)=\phi(t-k)$
- The approximation is represented by the multiplier coefficients of the translates $(\phi_k)_k$
- Mathematically, $x(t)\equiv \sum_ka_k\phi_k(t)$
Malleability Property
Dilatability of the building block to yield higher- or lower-resolution approximations
- A dilate of $\phi$ at scale is $\phi_{j,0}=2^{j\over 2}\phi(2^jt)$
- The translates of dilates are $\phi_{j,k}=2^{j\over 2}\phi(2^jt-k)$
- Note that the domain size of $\phi_{1,k}$ is half that of $\phi$ , and the domain size of $\phi_{2,k}$ is quarter that of $\phi$ , and so on. Thus the music terminology ``scale''
- The multiplier $2^{j\over 2}$ is to to make the dilates have the same energy (i.e., square integral) as $\phi$
- The higher the scale , the higher the resolution, because the domain size of the building block is finer
Analysis/Decomposition Property (part 1)
Ability to derive the lower-resolution coefficients from the next higher-resolution coefficients without any reference to the original function
Analysis/Decomposition Property (part 2)
Ability to derive the coefficients of the error (residual or difference) between a higher-resolution approximation and a lower-resolution approximation,
- using only the higher-resolution coefficients
- without any reference to the original function
- using a building block $\psi$ , called wavelet
- mathematically, $e(t)= \sum_kb_k\psi_k$
Synthesis Property
Ability to derive the higher-resolution coefficients directly from the lower resolution and the error, without any reference to the original function
Good-fit Property
Choice of a good building block $\phi$ for the application signal(s) so that the error between the and the approximations is as small as possible

3. Illustrations of Translates and Dilates

4. Mathematical Formulations

Denote by the approximation of at scale . That is,
- $x_j(t)=\sum_kx_k\phi_{j,k}(t)$ for some
- is the best approximation of among all linear combinations of $(\phi_{j,k}(t))_k$
Let $e_{j-1}(t)=x_j(t)-x_{j-1}(t)$ be the error between scale- and scale- approximations of ;

$\begin{displaymath}x_j(t)=x_{j-1}+e_{j-1}(t)\end{displaymath}$
Notation:

$x_j(t)=\sum_kx_k\phi_{j,k}(t)$

$x_{j-1}(t)=\sum_ku_k\phi_{j-1,k}(t)$

$e_{j-1}(t)=\sum_kv_k\psi_{j-1,k}(t)$
Desirable properties
- For every in , $x_j(t)\longrightarrow x(t)$ as $j\longrightarrow \infty$
  (convergence in )
- Analysis property: derivability of and from
- Synthesis property: derivability of from both and

5. Conditions for Achieving the Analysis and Synthesis Properties

Choose the scaling function $\phi$ to be composable from its own higher-resolution dilates & translates
Choose the wavelet $\psi$ to be composable from the higher-resolution dilates & translates of $\phi$
Finally, higher-resolution dilates-&-translates of $\phi$ should be decomposable into lower-resolution dilates-&-translates of $\phi$ and $\psi$
Mathematically
- $\phi(t)=\sum_np_n\phi(2t-n)$ , for some sequence
- $\psi(t)=\sum_nq_n\phi(2t-n)$ , for some sequence
- $\phi(2t-k)={1\over 2}\sum_n\left[ g_{2n-k}\phi(t-n)+h_{2n-k}\psi(t-n)\right]$ ,
  for some sequences and

6. Relation to Subband Coding

Theorem
Let
$x_j(t)=\sum_kx_k\phi_{j,k}(t)$
$x_{j-1}(t)=\sum_ku_k\phi_{j-1,k}(t)$
$e_{j-1}(t)=\sum_kv_k\psi_{j-1,k}(t)$
where
$\phi(t)=\sum_np_n\phi(2t-n)$
$\psi(t)=\sum_nq_n\phi(2t-n)$
$\phi(2t-k)={1\over 2}\sum_n\left[ g_{2n-k}\phi(t-n)+h_{2n-k}\psi(t-n)\right]$ .
Then is related with and by the following subband coder:

Proof of the Theorem
(Analysis stage: from to and )
- $\phi(2t-k)={1\over 2}\sum_n\left[ g_{2n-k}\phi(t-n)+h_{2n-k}\psi(t-n)\right]$
- $\phi_{j,k}(t)={1\over \sqrt{2}}\sum_n\left[ g_{2n-k}\phi_{j-1,n}(t)+h_{2n-k}\psi_{j-1,n}(t)\right]$
- On the other hand,
  $x_j(t)=x_{j-1}(t)+e_{j-1}(t)= \sum_nu_n\phi_{j-1,n}(t)+\sum_nv_n\psi_{j-1,n}(t)$
- Therefore,
  - $u_n=\sum_n{g_{2n-k}\over\sqrt{2}}x_k={\overline{u_{2n}}}$
  - $v_n=\sum_n{h_{2n-k}\over\sqrt{2}}x_k={\overline{v_{2n}}}$
- That is, is the down-sampled ${g\over\sqrt{2}}$ -filtered , and
  is the down-sampled ${h\over\sqrt{2}}$ -filtered
Proof of the Theorem
(Synthesis stage: from and to )
- $\phi(t)=\sum_np_n\phi(2t-n)$
- Similarly, $\psi_{j-1,n}=\sum_n{q_n\over\sqrt{2}}\phi_{j,2k+n}(t)$
- ```
(where )
```
- Since $x_j(t)=\sum_rx_r\phi_{j,r}(t)$ , it follows that
  
  $\begin{displaymath}x_r=\sum_k\left[{{p_{r-2k}}\over\sqrt{2}}u_k+{{q_{r-2k}}\over\sqrt{2}}v_k\right]\end{displaymath}$
  
  which is precisely the sum of the ${p\over\sqrt{2}}$ -filtered upsampled and the ${q\over\sqrt{2}}$ -filtered upsampled
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7. Illustration of Wavelets' Dynamic Adjustment to Regional Variations without Blockiness (Comparison with Whole DCT)

8. Mathematical Method for Computing the Four Filters

, , ,

(Symmetric Filters)

Define the -transforms of the four filters, with a slight scale modification:
$G(z)={1\over 2}\sum_kg_kz^k$ , $P(z)={1\over 2}\sum_kp_kz^k$ ,
$H(z)={1\over 2}\sum_kh_kz^k$ , $Q(z)={1\over 2}\sum_kq_kz^k$
The perfect reconstruction condition (seen before):
- :
- :
Take $H(z)=-z^{-1}P(-z)$ and , i.e.,

$\begin{displaymath}h_k=(-1)^kp_{k+1}\ \ and\ \ q_k=(-1)^kg_{k-1}\end{displaymath}$
That choice of and satisfies PR2 and makes PR1 equivalent to

$\begin{displaymath}PR'1:\ G(z)P(z)+G(-z)P(-z)=1\end{displaymath}$

Theorem
The symmetry of the filters along with implies that for any $z=e^{-i\omega}$

$P(z)=e^{-i{m\over 2}\omega}\cos^l({\omega\over 2}) S(\cos \omega)$

$G(z)=e^{i{m\over 2}\omega}\cos^{\hat l}({\omega\over 2}) {\hat S}(\cos \omega)$

for some integers $m,\ l$ and ${\hat l}$ , and some polynomials and ${\hat S}$ , such that $m,\ l$ and ${\hat l}$ have the same parity, and and ${\hat l}$ are positive.
Let $N={{l+{\hat l}}\over 2}$
Therefore,
- $G(z)P(z)=(\cos^2({\omega\over 2}))^N S(\cos \omega){\hat S}(\cos \omega)$
- $G(-z)P(-z)=(\sin^2({\omega\over 2}))^N S(-\cos \omega){\hat S}(-\cos \omega)$ ,
  because $-z=e^{-(\omega+\pi)}$
By letting $x=\sin^2({{\omega}\over 2})$ and defining the polynomial
$F(x)=S(\cos \omega){\hat S}(\cos \omega)=S(1-2x){\hat S}(1-2x)$ ,
one concludes from the previous step and the following equation

$\begin{displaymath}(1-x)^NF(x)+x^NF(1-x)=1\end{displaymath}$
The general solution of that equation is of the form

$\begin{displaymath}F(x)=R(x)+x^NT_0(x)\end{displaymath}$

where is a polynomial of degree satisfying

,
and is an arbitrary polynomial such that
The equation of implies that

$R(x)+x^N(1-x)^{-N}R(1-x)=(1-x)^{-N}.$

Since $(1-x)^{-N}=\sum_{k\ge 0}\left(\begin{array}{c} N-k+1\\ k \end{array}\right)x^k$ and is of degree , it follows that

$\begin{displaymath}R(x)=\sum_{k=0}^{N-1}\left(\begin{array}{c} N-k+1\\ k \end{array}\right)x^k\end{displaymath}$
Therefore,

$\begin{displaymath}F(x)=\sum_{k=0}^{N-1}\left(\begin{array}{c} N-k+1\\ k \end{array}\right)x^k+x^NT(\cos{\omega})\end{displaymath}$

where $T(\cos{\omega})=T_0(x)$ is an arbitrary odd polynomial.
In conclusion, to get $S(\cos{\omega})$ and ${\hat S}(\cos{\omega})$ , first factor by means of root finding, then give some factors to $S(\cos{\omega})$ and the remaining factors to ${\hat S}(\cos{\omega})$ .

9. Algorithm for Computing the Taps of the Filters

Input: specify , ${\hat l}$ , and the polynomial
$N={{l+{\hat l}}\over 2}$ and $F(x)=\sum_{k=0}^{N-1}\left(\begin{array}{c} N-k+1\\ k \end{array}\right)x^k+x^NT(1-2x)$
Find the roots of
Thus is factored into ,
where every is a linear or quadratic polynomial in
Input: specify the index set of the factors going to
$S(\cos{\omega}):=\prod_{k\in L}F_k(x)= \prod_{k\in L}F_k(-{{1-z}\over 2}^2z^{-1})$
${\hat S}(\cos{\omega}):=\prod_{{k\in{\overline L}}}F_k(x)= \prod_{{k\in{\overline L}}}F_k(-{{1-z}\over 2}^2z^{-1})$
Compute the coefficients of
$P(z)=z^{{m-l}\over 2}\left({{1+z}\over 2}\right)^l\prod_{k\in L}F_k(-{{1-z}\over 2}^2z^{-1})$
Let the coefficient of in , for all
Compute the coefficients of
$G(z)=(-1)^{{\hat l}}z^{-{{m+{\hat l}}\over 2}}\left({{1-z}\over 2}\right)^{{\hat l}}\prod_{{k\in{\overline L}}}F_k(-{{1-z}\over 2}^2z^{-1})$
Let the coefficient of in , for all
Normalize and so that $\sum_kp_k=2$ and $\sum_kg_k=2$
Compute $h_k=(-1)^kp_{k+1}$ and $q_k=(-1)^kg_{k-1}$

10. Examples of Four-Filter Sets

11. Daubechies Orthogonal Wavelets

In orthogonal wavelets the following holds:
- $h_k=(-1)^kp_{k+1}$
- $q_k=(-1)^kg_{k-1}$
Thus, one filter fully specifies all the four filters
Daubechies Orthogonal wavelets are the most popular orthogonal wavelets
However, they are not symmetric filters, and thus lead leas to what is known as phase distortion in compression.
Therefore, they are not widely used for image compression.

Wavelets

Abdou Youssef

Wavelet-Based Approximation: Introduction and Motivation

Some ``Desirables'' in Approximation Theory

Illustrations of Translates and Dilates

Mathematical Formulations

Conditions for Achieving the Analysis and Synthesis Properties

Relation to Subband Coding

Illustration of Wavelets' Dynamic Adjustment to Regional Variations without Blockiness (Comparison with Whole DCT)

Mathematical Method for Computing the Four Filters

Algorithm for Computing the Taps of the Filters

Examples of Four-Filter Sets

Daubechies Orthogonal Wavelets

1. Wavelet-Based Approximation: Introduction and Motivation

2. Some ``Desirables'' in Approximation Theory

Building Block Property

Malleability Property

Analysis/Decomposition Property (part 1)

Analysis/Decomposition Property (part 2)

Synthesis Property

Good-fit Property

3. Illustrations of Translates and Dilates

4. Mathematical Formulations

5. Conditions for Achieving the Analysis and Synthesis Properties

6. Relation to Subband Coding

Theorem

Proof of the Theorem

Proof of the Theorem

7. Illustration of Wavelets' Dynamic Adjustment to Regional Variations without Blockiness (Comparison with Whole DCT)

8. Mathematical Method for Computing the Four Filters

(Symmetric Filters)

Theorem

9. Algorithm for Computing the Taps of the Filters

10. Examples of Four-Filter Sets

11. Daubechies Orthogonal Wavelets