Parametric Bernstein polynomial for least squares design of 3-D wavelet filter banks

The design of nonseparable three-dimensional (3-D) biorthogonal wavelet filter banks is addressed in this paper. The sampling is on the face centered orthorhombic (FCO) lattice and the ideal low-pass filter's passband shape is the truncated octahedron (TRO). We introduce a 3-D parametric Bernstein polynomial that preserves biorthogonality and gives a good approximation to the TRO shape. Furthermore, filters with arbitrarily flat frequency response for giving regular wavelet systems are readily obtainable. The free parameters of the Bernstein polynomial can be chosen to sharpen the frequency response of the filter. A least squares approach is employed for the design of the parameters. The design process is efficient as it involves solving linear equations and is noniterative. This approach provides a trade-off mechanism between the sharpness of roll-off and the degree of flatness.