# Design Technique and Efficient Polyphase Implementation for 2D Elliptically Shaped FIR Filters

## TL;DR

This paper introduces a new method for designing 2D FIR filters with elliptical frequency responses, enabling efficient image processing and accurate reconstruction.

## Contribution

A novel analytical approach for designing elliptically shaped 2D FIR filters with directional selectivity and orientation control.

## Key findings

- The proposed filters achieve accurate shapes with negligible distortions.
- The polyphase implementation reduces computational complexity compared to direct filtering.
- The filters efficiently extract straight lines from images with specified orientations.

## Abstract

This paper presents a novel analytical approach for the efficient design of a particular class of 2D FIR filters, having a frequency response with an elliptically shaped support in the frequency plane. The filter design is based on a Gaussian shaped prototype filter, which is frequently used in signal and image processing. In order to express the Gaussian prototype frequency response as a trigonometric polynomial, we developed it into a Fourier series up to a specified order, given by the imposed approximation precision. We determined analytically a 1D to 2D frequency transformation, which was applied to the factored frequency response of the prototype, yielding directly the factored frequency response of a directional, elliptically shaped 2D filter, with specified selectivity and an orientation angle. The designed filters have accurate shapes and negligible distortions. We also designed a 2D uniform filter bank of elliptical filters, which was then applied in decomposing a test image into sub-band images, thus proving its usefulness as an analysis filter bank. Then, the original image was accurately reconstructed from its sub-band images. Very selective directional elliptical filters can be used in efficiently extracting straight lines with specified orientations from images, as shown in simulation examples. A computationally efficient implementation at the system level was also discussed, based on a polyphase and block filtering approach. The proposed implementation is illustrated for a smaller size of the filter kernel and input image and is shown to have reduced computational complexity due to its parallel structure, being much more arithmetically efficient compared not only to the direct filtering approach but also with the most recent similar implementations.