A convolution approach to the circle Hough transform for arbitrary radius

Abstract

The Hough transform is a well-established family of algorithms for locating and describing geometric figures in an image. However, the computational complexity of the algorithm used to calculate the transform is high when used to target complex objects. As a result, the use of the Hough transform to find objects more complex than lines is uncommon in real-time applications. We describe a convolution method for calculating the Hough transform for finding circles of arbitrary radius. The algorithm operates by performing a three-dimensional convolution of the input image with an appropriate Hough kernel. The use of the fast Fourier transform to calculate the convolution results in a Hough transform algorithm with reduced computational complexity and thus increased speed. Edge detection and other convolution-based image processing operations can be incorporated as part of the transform, which removes the need to perform them with a separate pre-processing or post-processing step. As the Discrete Fourier Transform implements circular convolution rather than linear convolution, consideration must be given to padding the input image before forming the Hough transform.