Wikipedia / Refinable function
Wikipedia / Refinable function
In mathematics, in the area of wavelet analysis, a refinable function is a function which fulfils some kind of self-similarity if
This condition is called refinement equation, dilation equation or two-scale equation.
Using the convolution 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.
is linear. A refinable function is an eigenfunction of that operator. Its absolute value is refinable, too.
These functions play a fundamental role in wavelet theory as scaling functions.
Properties
Values at integral points
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.
Using the discretization here, and the transfer matrix
This is again a fixed-point equation. But this one can now be considered as an eigenvector-eigenvalue has the eigenvalue1.
Values at dyadic points
.
The star denotes the convolution you get the values at all finer scales.
Convolution
.
Differentiation
. 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.
Integration
.
has bounded support, then we can interpret integration as convolution with the Heaviside function and apply the convolution law.
Scalar products
is the adjoint with respect to convolution is the flipped and complex conjugated.
, 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.
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.
minus logarithm of the spectral radius.
Generalization
. The most simple generalization is about tensor products.
The scheme can be generalized even more to different scaling factors with respect to different dimensions or even to mixing data between dimensions..)
Examples
If the definition is extended to distributions, then the Dirac impulse, that is known as Kronecker delta.
The Heaviside function.
The triangular function is a refinable function. B-spline functions with successive integral nodes are refinable, because of the convolution theorem and the refinability of the characteristic function (a boxcar function).
All polynomial functions.
A rational function is refinable if and only if it can be represented using partial fractions is a positive natural number is a real sequence with finitely many non-zero elements (a Laurent polynomial is the associated refinement mask.