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

More verbosely

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

PropertiesEdit

  •  .
  • 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   be the even-indexed coefficients of   ( ) and let   be the odd-indexed coefficients of   ( ).
Then  , where   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   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   is singular whenever   is singular. This property also tells, how vectors from the null space of   can be converted to null space vectors of  .
  • If   is an eigenvector of   with respect to the eigenvalue  , i.e.
 ,
then   is an eigenvector of   with respect to the same eigenvalue, i.e.
 .
  • Let   be the eigenvalues of  , which implies   and more generally  . This sum is useful for estimating the spectral radius of  . There is an alternative possibility for computing the sum of eigenvalue powers, which is faster for small  .
Let   be the periodization of   with respect to period  . That is   is a circular filter, which means that the component indexes are residue classes with respect to the modulus  . Then with the upsampling operator   it holds
 
Actually not   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  .

See alsoEdit

ReferencesEdit

  • Strang, Gilbert (1996). "Eigenvalues of   and convergence of the cascade algorithm". IEEE Transactions on Signal Processing. 44. pp. 233–238.
  • Thielemann, Henning (2006). Optimally matched wavelets (PhD thesis). (contains proofs of the above properties)