Generalization of Brady-Yong Algorithm for Fast Hough Transform to Arbitrary Image Size

TL;DR

A new algorithm for calculating the Hough (discrete Radon) transform for images of arbitrary size is presented that generalizes the Brady-Yong algorithm from which it inherits the optimal computational complexity.

Abstract

Nowadays, the Hough (discrete Radon) transform (HT/DRT) has proved to be an extremely powerful and widespread tool harnessed in a number of application areas, ranging from general image processing to X-ray computed tomography. Efficient utilization of the HT to solve applied problems demands its acceleration and increased accuracy. Along with this, most fast algorithms for computing the HT, especially the pioneering Brady-Yong algorithm, operate on power-of-two size input images and are not adapted for arbitrary size images. This paper presents a new algorithm for calculating the HT for images of arbitrary size. It generalizes the Brady-Yong algorithm from which it inherits the optimal computational complexity. Moreover, the algorithm allows to compute the HT with considerably higher accuracy compared to the existing algorithm. Herewith, the paper provides a theoretical analysis of the computational complexity and accuracy of the proposed algorithm. The conclusions of the performed experiments conform with the theoretical results.

Figures (2)

• Figure 1: (a) Experimentally assessed values of the $FHT2DS$ and $FHT2DT$ algorithms computational complexities normalized by $n^2 \log_2 n$. Dark violet points depict peak values of the $FHT2DT$ complexity. (b) Maximum orthotropic error of approximation of straight lines by $FHT2DS$ and $FHT2DT$ patterns normalized by $\log_2 n / 6$. Red points show the location of the highest local maxima of the normalized approximation error provided by $FHT2DS$ patterns. The green graph highlights the position of the derived in Theorem 2 estimation normalized by $\log_2 n / 6$.
• Figure 2: Measured values of the computational complexities of $FHT2DS$ and $FHT2DT$ algorithms divided by the corresponding estimating expressions in Theorem 1 (for different image sizes $n$). (a) The dynamics of individually normalized complexities over the full range $n \leq 4096$. (b) Normalized complexity values dynamics over the segment $n \leq 64$. Points and dashed lines mark image sizes for which algorithms normalized complexities equal 1.