Analytic Design Technique for 2D FIR Circular Filter Banks and Their Efficient Implementation Using Polyphase Approach

This paper proposes an analytical design procedure for 2D FIR circular filter banks and also a novel, computationally efficient implementation of the designed filter bank based on a polyphase structure and a block filtering approach. The component filters of the bank are designed in the frequency domain using a specific frequency transformation applied to a low-pass, band-pass and high-pass 1D prototype with a specified Gaussian shape and imposed specifications (peak frequency, bandwidth). The 1D prototype filter frequency response is derived in a closed form as a trigonometric polynomial with a specified order using Fourier series, and then it is factored. Since the design starts from a 1D prototype with a factored transfer function, the frequency response of the designed 2D filter bank components also results directly in a factored form. The designed filters have an accurate shape, with negligible distortions at a relatively low order. We present the design of two types of circular filter banks: uniform and non-uniform (dyadic). An example of image analysis with the uniform filter bank is also provided, showing that the original image can be accurately reconstructed from the sub-band images. The proposed implementation is presented for a simpler case, namely for a smaller size of the filter kernel and of the input image. Using the polyphase and block filtering approach, a convenient implementation at the system level is obtained for the designed 2D FIR filter, with a relatively low computational complexity.