Transfer matrix
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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 "":
- .
[edit] 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 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 it holds
- Actually not n − 2 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.
- From the previous statement we can derive an estimate of the spectral radius of . It holds
- where is the size of the filter and if all eigenvalues are real, it is also true that
- ,
- where .
[edit] References
- Gilbert Strang: Eigenvalues of and convergence of the cascade algorithm. IEEE Transactions on Signal Processing, 44:233-238, 1996.
- Henning Thielemann: Optimally matched wavelets, 2006 (contains proofs of the above properties)