Transfer matrix

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)_{j,k} = h_{2\cdot j-k}.

More verbosely


T_h =
\begin{pmatrix}
h_{a  } &         &         &         &         &   \\
h_{a%2B2} & h_{a%2B1} & h_{a  } &         &         &   \\
h_{a%2B4} & h_{a%2B3} & h_{a%2B2} & h_{a%2B1} & 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}.

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

T_h\cdot x = (h*x)\downarrow 2.

Properties

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%2B1}).
Then \det T_h = (-1)^{\lfloor\frac{b-a%2B1}{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.
\mathrm{tr}~T_{g*h} = \mathrm{tr}~T_{g} \cdot \mathrm{tr}~T_{h}
\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.
This is a concretion of the determinant property above. From the determinant property one knows that T_{g*h} is singular whenever T_{h} is singular. This property also tells, how vectors from the null space of T_{h} can be converted to null space vectors of T_{g*h}.
T_{h}\cdot x = \lambda\cdot x,
then x*(1,-1) is an eigenvector of T_{h*(1,1)} with respect to the same eigenvalue, i.e.
T_{h*(1,1)}\cdot (x*(1,-1)) = \lambda\cdot (x*(1,-1)).
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.
\varrho(T_h) \ge \frac{a}{\sqrt{\# h}} \ge \frac{1}{\sqrt{3\cdot \# h}}
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

References