Optimization of the Higher Density Discrete Wavelet Transform and of Its Dual Tree

We propose a parameterization of the finite-impulse-response (FIR) filters that generate the higher density discrete wavelet transform (HD-DWT). The parameterization has the form of a linear system whose variables are the coefficients of two positive trigonometric polynomials. The parameterization allows the easy separation of independent variables and can be used to transform the optimization of HD-DWT into semidefinite programming (SDP) form. Using the parameterization, we are able to optimize filters whose length is greater than the minimal one imposed by regularity constraints. We also propose two optimization methods for the dual-tree HD-DWT. The first generalizes the design based on allpass approximation of the half sample delay. The second is a brute force optimization based on generating values for the independent variables by using successive sections through the admissible set. As optimization criteria, we use an analyticity measure in the latter method and stopband energy in the former ones. For each type of design, we present examples and compare them with previous work.