Analytical design of 3-D wavelet filter banks using the multivariate Bernstein polynomial

The design of 3-D multirate filter banks where the downsampling/upsampling is on the FCO (face centered orthorhombic) lattice is addressed. With such a sampling lattice, the ideal 3-D subband of the low-pass filter is of the TRO (truncated octahedron) shape. The transformation of variables has been shown previously to be an effective technique for designing M-D filter banks. We present a design technique for the transformation function using the multivariate Bernstein polynomial which provides a good approximation to the TRO subband shape. The method is analytically based and does not require any optimization procedure. Closed form expressions are obtained for the filters of any order. Another advantage of this technique is that it yields filters with a flat frequency response at the aliasing frequency (ω₁,ω₂,ω₃ )=(π,π,π). The flatness is important for giving regular discrete wavelet transform systems