Two problems in low dimensional wavelet filter design

(Abstract)

We shall discuss two problems in low dimensional wavelet filter design. The first one is designing one-dimensional symmetric tight frame filters with rational coefficients. Filters whose coefficients are rational numbers, more specifically, dyadic numbers, have lots of advantages in computation like high computation speed and perfect numeric reconstruction etc. Hence, it is desirable to construct symmetric filters $b1(z)$, $b2(z)$ and $a(z)$ to have rational coefficients. We shall give a straight forward algorithm and a necessary and sufficient condition for this construction. A new example will be presentedto to demonstrate our algorithm.

Another problem is designing two-dimensional Hermite interpolating scheme of order two. Hermite interpolating scheme is useful in Computer Aimed Graphics Design(CAGD). Powell-Sabin scheme is a well-known bivariate polynomial Hermite scheme of order one. One of our result is that unlike Powell-Sabin scheme, there is no polynomial bivariate Hermite scheme of order two. Also, a non-polynomial bivariate Hermite scheme of order two will be given.