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

$\varphi (x)=2\cdot \sum _{k=0}^{N-1}h_{k}\cdot \varphi (2\cdot x-k)$

This condition is called refinement 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:

$\varphi =2\cdot D_{1/2}(h*\varphi )$

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 $\varphi \mapsto 2\cdot D_{1/2}(h*\varphi )$ 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 $c\cdot \varphi$ is refinable, too.

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

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Generalization

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

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 then one. (Maybe it also suffices that $|\det M|>1$.)

Formally the two scale equation does not change very much:

$\varphi (x)=|\det M|\cdot \sum _{k\in \mathbb {Z} ^{d}}h_{k}\cdot \varphi (M\cdot x-k)$

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

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