A High-Accuracy Fast Hough Transform with Linear-Log-Cubed Computational Complexity for Arbitrary-Shaped Images

The Hough transform (HT) is a fundamental tool across various domains, from classical image analysis to neural networks and tomography. Two key aspects of the algorithms for computing the HT are their computational complexity and accuracy - the latter often defined as the error of approximation of continuous lines by discrete ones within the image region. The fast HT (FHT) algorithms with optimal linearithmic complexity - such as the Brady-Yong algorithm for power-of-two-sized images - are well established. Generalizations like FHT2DT extend this efficiency to arbitrary image sizes, but with reduced accuracy that worsens with scale. Conversely, accurate HT algorithms achieve constant-bounded error but require near-cubic computational cost. This paper introduces FHT2SP algorithm - a fast and highly accurate HT algorithm. It builds on our development of Brady's superpixel concept, extending it to arbitrary shapes beyond the original power-of-two square constraint, and integrates it into the FHT2DT algorithm. With an appropriate choice of the superpixel's size, for an image of shape w × h, the FHT2SP algorithm achieves near-optimal computational complexity O(wh ln3 w), while keeping the approximation error bounded by a constant independent of image size, and controllable via a meta-parameter. We provide theoretical and experimental analyses of the algorithm's complexity and accuracy.