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

Abstract

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.

Analytical Design of Gaussian Anisotropic 2D FIR Filters and Their Implementation Using the Block Filtering Approach

“…The advantages of the proposed design technique for these 2D directional filters, which also apply to other types of 2D filters, as presented in [49][50][51][52][53][54], were highlighted in detail in Section 2.1. The most important one would be the flexibility in design due to the fact that the resulting 2D filters have a parametric, factored frequency response, which allows them to be easily tuned and adjusted according to given specifications.…”

“…A comparison can also be made with the first author's previous works approaching related design methods [49][50][51][52][53][54]. In [49], a class of adjustable Gaussian filters is proposed, but they are of IIR type.…”

“…Regarding the implementation part, the proposed polyphase block-filtering algorithm continues the algorithm from [54] but is superior in efficiency. It has a higher degree of parallelism and different decimation factors and, therefore, performs better regarding computational complexity, as discussed below.…”

“…Various analytic techniques for the design of 2D filters, in particular with directional characteristics, have been proposed by the first author in previous papers, such as parametric IIR filters for directional Gaussian smoothing [49]; directional and square-shaped 2D IIR filters based on digital prototypes [50,51]; directional Gaussian 2D FIR filters, separable along three axes [52]; and adjustable 2D filters with elliptical and circular symmetry [53]. A computationally efficient implementation for a 2D circular FIR filter bank, based on a polyphase decomposition approach, has been proposed in [54].…”