Minimax design of two-channel critically sampled graph QMF banks

Abstract

We address here the issue of designing high spectral selectivity graph filter banks, which require high-degree polynomial approximation. To achieve sharp transition band roll required for high-selectivity, the minimax criterion is used in the design formulation. A constraint optimization method is used for the design process, and a spectral transformation is employed to circumvent the numerical problems that arise with high-degree polynomials. Using the transformation, approximation problem can be recast as an approximation problem with trigonometric polynomials. The transformed constraint optimization problem can be solved using the readily available sequential quadratic programming (SQP) method. High order filters, with low maximum approximation error (MAE) and low maximum reconstruction error (MRE), can be designed with this method. An algorithm is also developed that allows the user to choose design parameters that will give the lowest possible MRE. The method can also be adapted to trade-off between reconstruction quality and spectral selectivity of the resulting filter bank. The suite of techniques presented allow the user to easily shape the spectral response of the filters, to suit the application at hand. Numerous examples, with comparisons, will be presented to demonstrate the versatility of the proposed techniques. Applications to nonlinear approximation and denoising of graph signal will also be considered.