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In mathematics, in the area of wavelet analysis, a 'refinable function' is a function which fulfills some kind of self-similarity. A function is called refinable with respect to the mask $h$ if
:
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:
:
★ arphi)
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
★ arphi) is linear.
A refinable function is an eigenfunction of that operator.
Its absolute value is not defined.
That is, if is a refinable function,
then for every $c$ the function is refinable, too.
These functions play a fundamental role in wavelet theory as scaling functions.

Contents

Properties

Values at integral points

Values at dyadic points

Convolution

Scalar products

Smoothness

Generalization

References

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 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
:$$
egin{pmatrix}
arphi(a)\
arphi(a+1)\
dots\
arphi(b)
end{pmatrix}
=
egin{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
egin{pmatrix}
arphi(a)\
arphi(a+1)\
dots\
arphi(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
:.
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.

Values at dyadic points

From the values at integral points you can derive the values at dyadic points,
i.e. points of the form $kcdot\; 2^\{-j\}$, with $kinmathbb\{Z\}$ and $jinmathbb\{N\}$.
:
★ arphi))
:
★ arphi)
:
★ arphi)) = 2cdot (h
★ Qarphi)
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 .
By replacing iteratedly by you get the values at all finer scales.
:
★ Q(D_{2^j}arphi))

Convolution

If is refinable with respect to $h$,
and $psi$ is refinable with respect to $g$,
then
★ 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
:
angle = langle arphi
★ psi^
★ , T_kdelta
angle = (arphi
★ psi^
★ )(k)
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. $psi^$
★ (t) = overline{psi(-t)}.
Because of the above property,
★ 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 $(1,1)$ (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 $T\_\{q$
★ q^
★ }.

Generalization

The concept of refinable functions can be generalized to functions of more than one variable,
that is functions from $R^d\; o\; R$.
The most simple generalization is about tensor products.
If and $psi$
are refinable with respect to $h$ and $g$, respectively,
then
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 then one.
(Maybe it also suffices that $|det\; M|>1$.)
Formally the two scale equation does not change very much:
:
:
★ arphi)

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