Transfer matrix

The transfer matrix appears in the treatment of refinable functions.

For the mask $h$, which is vector with component indexes from $a$ to $b$, the transfer matrix of $h$, we call it $T_{h}$ here, is defined as

$(T_{h})_{j,k}=h_{2\cdot j-k}$.

More verbosely

$T_{h}={\begin{pmatrix}h_{a}&&&&&\\h_{a+2}&h_{a+1}&h_{a}&&&\\h_{a+4}&h_{a+3}&h_{a+2}&h_{a+1}&h_{a}&\\\ddots &\ddots &\ddots &\ddots &\ddots &\ddots \\&h_{b}&h_{b-1}&h_{b-2}&h_{b-3}&h_{b-4}\\&&&h_{b}&h_{b-1}&h_{b-2}\\&&&&&h_{b}\end{pmatrix}}$

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 $h_{\mathrm {e} }$ be the even indexed coefficients of $h$ ($(h_{\mathrm {e} })_{k}=h_{2\cdot k}$) and let $h_{\mathrm {o} }$ be the odd indexed coefficients of $h$ ($(h_{\mathrm {o} })_{k}=h_{2\cdot k+1}$).

Then $\det T_{h}=(-1)^{\lfloor {\frac {b-a+1}{4}}\rfloor }\cdot h_{a}\cdot h_{b}\cdot \mathrm {res} (h_{\mathrm {e} },h_{\mathrm {o} })$, where $\mathrm {res}$ is the resultant.

This connection allows for fast computation using the Euclidean algorithm.

• For the trace of the transfer matrix of convolved masks holds

$\mathrm {tr} ~T_{g*h}=\mathrm {tr} ~T_{g}\cdot \mathrm {tr} ~T_{h}$

• For the determinant of the transfer matrix of convolved mask holds

$\det T_{g*h}=\det T_{g}\cdot \det T_{h}\cdot \mathrm {res} (g_{-},h)$

where $g_{-}$ denotes the mask with alternating signs, i.e. $(g_{-})_{k}=(-1)^{k}\cdot g_{k}$.

• Let $\lambda _{a},\dots ,\lambda _{b}$ be the eigenvalues of $T_{h}$, which implies $\lambda _{a}+\dots +\lambda _{b}=\mathrm {tr} ~T_{h}$ and more generally $\lambda _{a}^{n}+\dots +\lambda _{b}^{n}=\mathrm {tr} (T_{h}^{n})$. This sum is useful for estimating the spectral radius of $T_{h}$. There is an alternative possibility for computing the sum of eigenvalue powers, which is faster for small $n$.

Let $C_{k}h$ be the periodization of $h$ with respect to period $2^{k}-1$. That is $C_{k}h$ is a circular filter, which means that the component indexes are residue classes with respect to the modulus $2^{k}-1$. Then with the upsampling operator $\uparrow$ it holds

$\mathrm {tr} (T_{h}^{n})=\left(C_{k}h*(C_{k}h\uparrow 2)*(C_{k}h\uparrow 2^{2})*\cdots *(C_{k}h\uparrow 2^{n-1})\right)_{[0]_{2^{n}-1}}$

Actually not $n-2$ convolutions are necessary, but only $2\cdot \log _{2}n$ ones, when applying the strategy of efficient computation of powers. Even more the approach can be further sped up using the Fast Fourier transform.

References

• Henning Thielemann: Optimally matched wavelets (contains proofs of the above properties)