Transfer matrix

This article is about the transfer matrix in wavelet theory. For the transfer matrix method in statistical physics, see Transfer-matrix method. For the transfer matrix method in optics, see Transfer-matrix method (optics).

In applied mathematics, the transfer matrix is a formulation in terms of a block-Toeplitz matrix of the two-scale equation, which characterizes refinable functions. Refinable functions play an important role in wavelet theory and finite element theory.

For the mask , which is a vector with component indexes from to , the transfer matrix of , we call it here, is defined as

More verbosely

The effect of can be expressed in terms of the downsampling operator "":


More precisely:
Let be the even-indexed coefficients of () and let be the odd-indexed coefficients of ().
Then , where is the resultant.
This connection allows for fast computation using the Euclidean algorithm.
where denotes the mask with alternating signs, i.e. .
This is a concretion of the determinant property above. From the determinant property one knows that is singular whenever is singular. This property also tells, how vectors from the null space of can be converted to null space vectors of .
then is an eigenvector of with respect to the same eigenvalue, i.e.
Let be the periodization of with respect to period . That is is a circular filter, which means that the component indexes are residue classes with respect to the modulus . Then with the upsampling operator it holds
Actually not convolutions are necessary, but only ones, when applying the strategy of efficient computation of powers. Even more the approach can be further sped up using the Fast Fourier transform.
where is the size of the filter and if all eigenvalues are real, it is also true that
where .

See also


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