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Wavelet and scaling functions

`[PHI,PSI,XVAL] = wavefun(`

* 'wname'*,ITER)

[PHI1,PSI1,PHI2,PSI2,XVAL] = wavefun(

`'wname'`

[PHI,PSI,XVAL] = wavefun(

`'wname'`

[PSI,XVAL] = wavefun(

`'wname'`

[...] = wavefun(

`wname`

[...] = wavefun(

`'wname'`

[...] = wavefun(

`'wname'`

[...] = wavefun(

`'wname'`

[...] = wavefun(

`'wname'`

[...] = wavefun(

`'wname'`

The function `wavefun`

returns
approximations of the wavelet function * 'wname'* and
the associated scaling function, if it exists. The positive integer

`ITER`

determines
the number of iterations computed; thus, the refinement of the approximations.*For an orthogonal wavelet*:

`[PHI,PSI,XVAL] = wavefun(`

returns
the scaling and wavelet functions on the points grid * 'wname'*,ITER)

`XVAL`

. *For a biorthogonal wavelet*:

`[PHI1,PSI1,PHI2,PSI2,XVAL] = wavefun(`

returns
the scaling and wavelet functions both for decomposition * 'wname'*,ITER)

`(PHI1,PSI1)`

and
for reconstruction `(PHI2,PSI2)`

. *For a Meyer wavelet*:

`[PHI,PSI,XVAL] = wavefun(`

* 'wname'*,ITER)

*For a wavelet without scaling function (e.g., Morlet,
Mexican Hat, Gaussian derivatives wavelets or complex wavelets)*:

`[PSI,XVAL] = wavefun(`

* 'wname'*,ITER)

`[...] = wavefun(`

,
where * wname*,A,B)

`A`

and `B`

are positive integers,
is equivalent to `[...] = wavefun(``'wname'`

,max(A,B))

,
and draws plots. When `A`

is set equal to the special value
0,

`[...] = wavefun(`

is equivalent to,0)`'wname'`

`[...] = wavefun(`

.,8,0)`'wname'`

`[...] = wavefun(`

is equivalent to)`'wname'`

`[...] = wavefun(`

.,8)`'wname'`

The output arguments are optional.

On the following graph, 10 piecewise linear approximations of
the `sym4`

wavelet obtained after each iteration
of the cascade algorithm are shown.

% Set number of iterations and wavelet name. iter = 10; wav = 'sym4'; % Compute approximations of the wavelet function using the % cascade algorithm. for i = 1:iter [phi,psi,xval] = wavefun(wav,i); plot(xval,psi); hold on end title(['Approximations of the wavelet ',wav, ... ' for 1 to ',num2str(iter),' iterations']); hold off

For compactly supported wavelets defined by filters, in general no closed form analytic formula exists.

The algorithm used is the cascade algorithm. It uses the single-level inverse wavelet transform repeatedly.

Let us begin with the scaling function ϕ.

Since ϕ is also equal to ϕ_{0,0},
this function is characterized by the following coefficients in the
orthogonal framework:

<ϕ, ϕ

> = 1 only if_{0,n}*n*= 0 and equal to 0 otherwise<ϕ, ψ

> = 0 for positive_{−j,k}*j*, and all*k*.

This expansion can be viewed as a wavelet decomposition structure. Detail coefficients are all zeros and approximation coefficients are all zeros except one equal to 1.

Then we use the reconstruction algorithm to approximate the function ϕ over a dyadic grid, according to the following result:

For any dyadic rational of the form *x* = *n*2^{−j} in
which the function is continuous and where *j* is
sufficiently large, we have pointwise convergence and

where *C* is a constant, and α is a positive
constant depending on the wavelet regularity.

Then using a good approximation of ϕ on dyadic rationals, we can use piecewise constant or piecewise linear interpolations η on dyadic intervals, for which uniform convergence occurs with similar exponential rate:

So using a *J*-step reconstruction scheme,
we obtain an approximation that converges exponentially towards ϕ
when *J* goes to infinity.

Approximations are computed over a grid of dyadic rationals covering the support of the function to be approximated.

Since a scaled version of the wavelet function ψ can also
be expanded on the (ϕ_{−1,n)})* _{n}*,
the same scheme can be used, after a single-level reconstruction starting
with the appropriate wavelet decomposition structure. Approximation
coefficients are all zeros and detail coefficients are all zeros except
one equal to 1.

For biorthogonal wavelets, the same ideas can be applied on each of the two multiresolution schemes in duality.

This algorithm may diverge if the function to be approximated is not continuous on dyadic rationals.

Daubechies, I., *Ten lectures on wavelets*,
CBMS, SIAM, 1992, pp. 202Äì213.

Strang, G.; T. Nguyen (1996), *Wavelets and Filter
Banks*, Wellesley-Cambridge Press.

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