Definitions

# Refinable function

In mathematics, in the area of wavelet analysis, a refinable function is a function which fulfills some kind of self-similarity. A function $varphi$ is called refinable with respect to the mask $h$ if
$varphi\left(x\right)=2cdotsum_\left\{k=0\right\}^\left\{N-1\right\} h_kcdotvarphi\left(2cdot x-k\right)$
This condition is called refinement equation, dilation equation or two-scale equation.

Using the convolution * of a function with a discrete mask and the dilation operator $D$ you can write more concisely:

$varphi=2cdot D_\left\{1/2\right\} \left(h * varphi\right)$
It means that you obtain the function, again, if you convolve the function with a discrete mask and then scale it back. There is an obvious similarity to iterated function systems and de Rham curves.

The operator $varphimapsto 2cdot D_\left\{1/2\right\} \left(h * varphi\right)$ is linear. A refinable function is an eigenfunction of that operator. Its absolute value is not defined. That is, if $varphi$ is a refinable function, then for every $c$ the function $ccdotvarphi$ is refinable, too.

These functions play a fundamental role in wavelet theory as scaling functions.

## Properties

### Values at integral points

A refinable function is defined only implicitly. It may also be that there are several functions which are refinable with respect to the same mask. If $varphi$ shall have finite support and the function values at integer arguments are wanted, then the two scale equation becomes a system of simultaneous linear equations.

Let $a$ be the minimum index and $b$ be the maximum index of non-zero elements of $h$, then one obtains


begin{pmatrix} varphi(a) varphi(a+1) vdots varphi(b) end{pmatrix} = begin{pmatrix} h_{a } & & & & & h_{a+2} & h_{a+1} & h_{a } & & & h_{a+4} & h_{a+3} & h_{a+2} & h_{a+1} & h_{a } & ddots & ddots & ddots & ddots & ddots & ddots & h_{b } & h_{b-1} & h_{b-2} & h_{b-3} & h_{b-4} & & & h_{b } & h_{b-1} & h_{b-2} & & & & & h_{b } end{pmatrix} cdot begin{pmatrix} varphi(a) varphi(a+1) vdots varphi(b) end{pmatrix} . Using the discretization operator, call it $Q$ here, and the transfer matrix of $h$, named $T_h$, this can be written concisely as
$Qvarphi = T_h cdot Qvarphi$.

This is again a fixed-point equation. But this one can now be considered as an eigenvector-eigenvalue problem. That is, a finitely supported refinable function exists only (but not necessarily), if $T_h$ has the eigenvalue 1.

From the values at integral points you can derive the values at dyadic points, i.e. points of the form $kcdot 2^\left\{-j\right\}$, with $kinmathbb\left\{Z\right\}$ and $jinmathbb\left\{N\right\}$.

$varphi = D_\left\{1/2\right\} \left(2cdot \left(h * varphi\right)\right)$
$D_2 varphi = 2cdot \left(h * varphi\right)$
$Q\left(D_2 varphi\right) = Q\left(2cdot \left(h * varphi\right)\right) = 2cdot \left(h * Qvarphi\right)$
The star denotes the convolution of a discrete filter with a function. With this step you can compute the values at points of the form $frac\left\{k\right\}\left\{2\right\}$. By replacing iteratedly $varphi$ by $D_2 varphi$ you get the values at all finer scales.
$Q\left(D_\left\{2^\left\{j+1\right\}\right\}varphi\right) = 2cdot \left(h * Q\left(D_\left\{2^j\right\}varphi\right)\right)$

### Convolution

If $varphi$ is refinable with respect to $h$, and $psi$ is refinable with respect to $g$, then $varphi*psi$ is refinable with respect to $h*g$.

### Scalar products

Computing the scalar products of two refinable functions and their translates can be broken down to the two above properties. Let $T$ be the translation operator. It holds

$langle varphi, T_k psirangle = langle varphi * psi^*, T_kdeltarangle = \left(varphi*psi^*\right)\left(k\right)$
where $psi^*$ is the adjoint of $psi$ with respect to convolution, i.e. $psi^*$ is the flipped and complex conjugated version of $psi$, i.e. $psi^*\left(t\right) = overline\left\{psi\left(-t\right)\right\}$.

Because of the above property, $varphi*psi^*$ is refinable with respect to $h*g^*$, and its values at integral arguments can be computed as eigenvectors of the transfer matrix.

### Smoothness

A refinable function usually has a fractal shape. The design of continuous or smooth refinable functions is not obvious. Before dealing with forcing smoothness it is necessary to measure smoothness of refinable functions. Using the Villemoes machine one can compute the smoothness of refinable functions in terms of Sobolev exponents.

In a first step the refinement mask $h$ is divided into a filter $b$, which is a power of the smoothness factor $\left(1,1\right)$ (this is a binomial mask) and a rest $q$. Roughly spoken, the binomial mask $b$ makes smoothness and $q$ represents a fractal component, which reduces smoothness again. Now the Sobolev exponent is roughly the order of $b$ minus logarithm of the spectral radius of $T_\left\{q*q^*\right\}$.

## Generalization

The concept of refinable functions can be generalized to functions of more than one variable, that is functions from $R^d to R$. The most simple generalization is about tensor products. If $varphi$ and $psi$ are refinable with respect to $h$ and $g$, respectively, then $varphiotimespsi$ is refinable with respect to $hotimes g$.

The scheme can be generalized even more to different scaling factors with respect to different dimensions or even to mixing data between dimensions. Instead of scaling by scalar factor like 2 the signal the coordinates are transformed by a matrix $M$ of integers. In order to let the scheme work, the absolute values of all eigenvalues of $M$ must be larger than one. (Maybe it also suffices that $|det M|>1$.)

Formally the two-scale equation does not change very much:

$varphi\left(x\right)=|det M|cdotsum_\left\{kinZ^d\right\} h_kcdotvarphi\left(Mcdot x-k\right)$

$varphi=|det M|cdot D_\left\{M^\left\{-1\right\}\right\} \left(h * varphi\right)$

## References

• Wolfgang Dahmen and Charles A. Micchelli: Using the refinement equation for evaluating integrals of wavelets. SIAM Journal Numerical Analysis, 30:507--537, 1993.
• Marc A. Berger and Yang Wang: Multidimensional two-scale dilation equations. In Charles K. Chui, editor, Wavelets: A Tutorial in Theory and Applications, volume 2 of Wavelet Analysis and its Applications, chapter IV, pages 295--323. Academic Press, Inc., 1992.
• Lars Villemoes: Sobolev regularity of wavelets and stability of iterated filter banks
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