Transfer Matrix

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 $h$, which is a vector with component indexes from $a$ to $b$,
the transfer matrix of $h$, we call it $T_h$ here, is defined as
:$(T_h)_ = h_.$
More verbosely
:$T_h = \begin h_ & & & & & \\ h_ & h_ & h_ & & & \\ h_ & h_ & h_ & h_ & h_ & \\ \ddots & \ddots & \ddots & \ddots & \ddots & \ddots \\ & h_ & h_ & h_ & h_ & h_ \\ & & & h_ & h_ & h_ \\ & & & & & h_ \end.$
The effect of $T_h$ can be expressed in terms of the downsampling operator "$\downarrow$":
:$T_h\cdot x = (h*x)\downarrow 2.$

Properties

* $T_h\cdot x = T_x\cdot h$.
* 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 $h_$ be the even-indexed coefficients of $h$ ($(h_)_k = h_$) and let $h_$ be the odd-indexed coefficients of $h$ ($(h_)_k = h_$).
:Then $\det T_h = (-1)^\cdot h_a\cdot h_b\cdot\mathrm(h_,h_)$, where $\mathrm$ is the resultant.
:This connection allows for fast computation using the Euclidean algorithm.
* For the trace of the transfer matrix of convolved masks holds
:$\mathrm~T_ = \mathrm~T_g \cdot \mathrm~T_h$
* For the determinant of the transfer matrix of convolved mask holds
:$\det T_ = \det T_g \cdot \det T_h \cdot \mathrm(g_-,h)$
:where $g_-$ denotes the mask with alternating signs, i.e. $(g_-)_k = (-1)^k \cdot g_k$.
* If $T_\cdot x = 0$, then $T_\cdot (g_-*x) = 0$.
: This is a concretion of the determinant property above. From the determinant property one knows that $T_$ is singular whenever $T_$ is singular. This property also tells, how vectors from the null space of $T_$ can be converted to null space vectors of $T_$.
* If $x$ is an eigenvector of $T_$ with respect to the eigenvalue $\lambda$, i.e.
: $T_\cdot x = \lambda\cdot x$,
:then $x*(1,-1)$ is an eigenvector of $T_$ with respect to the same eigenvalue, i.e.
: $T_\cdot (x*(1,-1)) = \lambda\cdot (x*(1,-1))$.
* Let $\lambda_a,\dots,\lambda_b$ be the eigenvalues of $T_h$, which implies $\lambda_a+\dots+\lambda_b = \mathrm~T_h$ and more generally $\lambda_a^n+\dots+\lambda_b^n = \mathrm(T_h^n)$. 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
:$\mathrm(T_h^n) = \left(C_k h * (C_k h\uparrow 2) * (C_k h\uparrow 2^2) * \cdots * (C_k h\uparrow 2^)\right)_$
: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.
* From the previous statement we can derive an estimate of the spectral radius of $\varrho(T_h)$. It holds
:$\varrho(T_h) \ge \frac \ge \frac$
:where $\# h$ is the size of the filter and if all eigenvalues are real, it is also true that
:$\varrho(T_h) \le a$,
:where $a = \Vert C_2 h \Vert_2$.

See also

* Hurwitz determinant

References

*
* {{cite thesis
, first=Henning
, last=Thielemann
, url=http://nbn-resolving.de/urn:nbn:de:gbv:46-diss000103131
, title=Optimally matched wavelets
, type=PhD thesis
, year=2006
(contains proofs of the above properties)

Wavelets
Numerical analysis