In mathematics, in the area of wavelet analysis, a refinable function is a function which fulfils some kind of self-similarity.
is called refinable with respect to the mask
Using the convolution (denoted by a star, *) of a function with a discrete mask and the dilation operator
one can write more concisely: It means that one obtains the function, again, if you convolve the function with a discrete mask and then scale it back.
There is a similarity to iterated function systems and de Rham curves.
A refinable function is an eigenfunction of that operator.
Its absolute value is not uniquely defined.
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.
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.
be the maximum index of non-zero elements of
That is, a finitely supported refinable function exists only (but not necessarily), if
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
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.
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.
, 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.
Using the Villemoes machine[2] one can compute the smoothness of refinable functions in terms of Sobolev exponents.
Roughly spoken, the binomial mask
represents a fractal component, which reduces smoothness again.
Now the Sobolev exponent is roughly the order of
minus logarithm of the spectral radius of
The most simple generalization is about tensor products.
[3] Instead of scaling by scalar factor like 2 the signal the coordinates are transformed by a matrix
In order to let the scheme work, the absolute values of all eigenvalues of
Formally the two-scale equation does not change very much: