Refinable function

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 ${\displaystyle \varphi }$ 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 ${\displaystyle a}$ be the minimum index and ${\displaystyle b}$ be the maximum index of non-zero elements of ${\displaystyle h}$, then one obtains

$${\displaystyle {\begin{pmatrix}\varphi (a)\\\varphi (a+1)\\\vdots \\\varphi (b)\end{pmatrix}}={\begin{pmatrix}h_{a}&&&&&\\h_{a+2}&h_{a+1}&h_{a}&&&\\h_{a+4}&h_{a+3}&h_{a+2}&h_{a+1}&h_{a}&\\\ddots &\ddots &\ddots &\ddots &\ddots &\ddots \\&h_{b}&h_{b-1}&h_{b-2}&h_{b-3}&h_{b-4}\\&&&h_{b}&h_{b-1}&h_{b-2}\\&&&&&h_{b}\end{pmatrix}}{\begin{pmatrix}\varphi (a)\\\varphi (a+1)\\\vdots \\\varphi (b)\end{pmatrix}}.}$$

Using the discretization operator, call it ${\displaystyle Q}$ here, and the transfer matrix of ${\displaystyle h}$, named ${\displaystyle T_{h}}$, this can be written concisely as

$${\displaystyle Q\varphi =T_{h}Q\varphi .}$$

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 ${\displaystyle T_{h}}$ has the eigenvalue 1.

From the values at integral points you can derive the values at dyadic points, i.e. points of the form ${\displaystyle k\cdot 2^{-j}}$, with ${\displaystyle k\in \mathbb {Z} }$ and ${\displaystyle j\in \mathbb {N} }$.

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

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

${\displaystyle Q(D_{2}\varphi )=Q(2\cdot (h*\varphi ))=2\cdot (h*Q\varphi )}$

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 ${\displaystyle {\frac {k}{2}}}$. By replacing iteratedly ${\displaystyle \varphi }$ by ${\displaystyle D_{2}\varphi }$ you get the values at all finer scales.

$${\displaystyle Q(D_{2^{j+1}}\varphi )=2\cdot (h*Q(D_{2^{j}}\varphi ))}$$

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

If ${\displaystyle \varphi }$ is refinable with respect to ${\displaystyle h}$, and the derivative ${\displaystyle \varphi '}$ exists, then ${\displaystyle \varphi '}$ is refinable with respect to ${\displaystyle 2\cdot h}$. This can be interpreted as a special case of the convolution property, where one of the convolution operands is a derivative of the Dirac impulse.

If ${\displaystyle \varphi }$ is refinable with respect to ${\displaystyle h}$, and there is an antiderivative ${\displaystyle \Phi }$ with ${\textstyle \Phi (t)=\int _{0}^{t}\varphi (\tau )\,\mathrm {d} \tau }$, then the antiderivative ${\displaystyle t\mapsto \Phi (t)+c}$ is refinable with respect to mask ${\textstyle {\frac {1}{2}}\cdot h}$ where the constant ${\displaystyle c}$ must fulfill ${\textstyle c\cdot \left(1-\sum _{j}h_{j}\right)=\sum _{j}h_{j}\cdot \Phi (-j)}$.

If ${\displaystyle \varphi }$ has bounded support, then we can interpret integration as convolution with the Heaviside function and apply the convolution law.

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

$${\displaystyle \langle \varphi ,T_{k}\psi \rangle =\langle \varphi *\psi ^{*},T_{k}\delta \rangle =(\varphi *\psi ^{*})(k)}$$

where ${\displaystyle \psi ^{*}}$ is the adjoint of ${\displaystyle \psi }$ with respect to convolution, i.e., ${\displaystyle \psi ^{*}}$ is the flipped and complex conjugated version of ${\displaystyle \psi }$, i.e., ${\displaystyle \psi ^{*}(t)={\overline {\psi (-t)}}}$.

Because of the above property, ${\displaystyle \varphi *\psi ^{*}}$ is refinable with respect to ${\displaystyle h*g^{*}}$, and its values at integral arguments can be computed as eigenvectors of the transfer matrix. This idea can be easily generalized to integrals of products of more than two refinable functions.[1]

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[2] one can compute the smoothness of refinable functions in terms of Sobolev exponents.

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