transfer matrix

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transfer matrix

Written by Eng.Admin

Tuesday, 04 November 2008 23:43

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

.
More verbosely

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

.
Properties

.
If you drop the first and the last column and move the odd indexed columns to the left and the even indexed columns to the right, then you obtain a transposed Sylvester matrix.
The determinant of a transfer matrix is essentially a resultant.

More precisely: Let he be the even indexed coefficients of h (

) and let ho be the odd indexed coefficients of h (

). Then

, where res is the resultant. This connection allows for fast computation using the Euclidean algorithm.

For the trace of the transfer matrix of convolved masks holds

For the determinant of the transfer matrix of convolved mask holds

where g − denotes the mask with alternating signs, i.e.

If , then .

This is a concretion of the determinant property above. From the determinant property one knows that Tg * h is singular whenever Th is singular. This property also tells, how vectors from the null space of Th can be converted to null space vectors of Tg * h.

If x is an eigenvector of Th with respect to the eigenvalue λ, i.e.

, then x * (1, − 1) is an eigenvector of Th * (1,1) with respect to the same eigenvalue, i.e.

Let be the eigenvalues of Th, which implies and more generally . This sum is useful for estimating the spectral radius of Th. There is an alternative possibility for computing the sum of eigenvalue powers, which is faster for small n.

Let Ckh be the periodization of h with respect to period 2k − 1. That is Ckh is a circular filter, which means that the component indexes are residue classes with respect to the modulus 2k − 1. Then with the upsampling operator