Smooth Wavelet Tight Frames with Zero Moments

Example 2: K0 = 5, K1 = K2 = 2

There are 4 distinct minimal-length solutions. The solutions in terms of radicals, obtained using Gröbner bases, are tabulated in the MATLAB program k522.m. The numerical solutions are tabulated in the file k522.float. The system k522(1) is shown in this figure. All four solutions are shown together on another page.

The filters h0, h1, h2 were found by converting the nonlinear design equations (eqs) into a degree Gröbner basis (gb.dp), converting that into a lexical Gröbner basis (gb.lp), and factorizing that into two Gröbner bases (gb.lp.fact.1, gb.lp.fact.2). Then the ordering of the variables was changed, to obtain two lexical Gröbner bases which are more compact (gb.lp.1, gb.lp.2). (However, if this ordering is used from the beginning, then the original lexical Gröbner basis does not factor.) All minimal-length pairs of scaling filters can be found by solving these 2 Gröbner bases.

The Gröbner bases indicate that there are 32 solutions to the nonlinear design equations (16 solutions from gb.lp.1 and 16 solutions from gb.lp.2). However, excluding time-reversal and negation, there are 4 distinct solutions (2 solutions from gb.lp.1 and 8 solutions from gb.lp.2). The solutions derived from gb.lp.1 and gb.lp.2 are tabulated respectively in the MATLAB programs c1.m and c2.m.

We also provide for this example the programs for reproducing the filters: the Maple program (setup) for generating the equations, the Singular program (sfile) for obtaining the Gröbner bases, and the Maple program (result) for solving the Gröbner bases. Executing these programs in sequence will regenerate the Gröbner bases and create two MATLAB programs c1.m and c2.m containing the solutions. We combined c1.m and c2.m into k522.m.

K0 = 5, K1 = K2 = 2. Solution 1.

Image of scaling/wavelet functions/filters.

The scaling and wavelet functions are on the right. The scaling and wavelet filters are on the left.