Design and Polyphase Implementation of Rotationally Invariant 2D FIR Filter Banks Based on Maximally Flat Prototype

Abstract

This paper presents a design approach for a class of rotationally invariant 2D filters of finite impulse response (FIR) type, which may form circular filter banks with imposed specifications. The design is conducted analytically in the frequency domain and starts from a maximally flat low-pass prototype based on a trapezoidal function with specified width and slope. Its trigonometric approximation is derived using the Fourier series expressed analytically, truncated to a number of terms depending on the imposed accuracy. The chosen trapezoidal function leads to significantly smaller ringing oscillations compared to the approximation of an ideal square characteristic. By shifting the LP prototype to various frequencies, the desired filter bank is generated, where the component filters have a specified bandwidth, steepness, and overlap. The 2D circular filter bank results by applying a specific frequency mapping to the factored frequency response of the prototype filter. Thus, the frequency responses of the 2D filter bank components will also result in factored form, which is an advantage in implementation. The circular filter bank is designed in two versions, a uniform and a non-uniform (dyadic) filter bank. The designed filter banks have accurate shapes and relatively low order for the specified parameters. These filter banks are then used in a sub-band image decomposition application. Finally, an efficient implementation of these filters at the system level is proposed based on polyphase decomposition and the block filtering technique with a high degree of parallelism, resulting in a lower computational complexity.