Simon Karpenko

Gleb Smirnov, Simon Karpenko

Fast Hough transform is a widely used algorithm in pattern recognition. The algorithm relies on approximating lines using a specific discrete line model called dyadic lines. The worst-case deviation of a dyadic line from the ideal line it used to construct grows as $O(log(n))$, where $n$ is the linear size of the image. But few lines actually reach the worst-case bound. The present paper addresses a statistical analysis of the deviation of a dyadic line from its ideal counterpart. Specifically, our findings show that the mean deviation is zero, and the variance grows as $O(log(n))$. As $n$ increases, the distribution of these (suitably normalized) deviations converges towards a normal distribution with zero mean and a small variance. This limiting result makes an essential use of ergodic theory.

E. I. Ershov, S. M. Karpenko

Hough transform is a popular low-level computer vision algorithm. Its computationally effective modification, Fast Hough transform (FHT), makes use of special subsets of image matrix to approximate geometric lines on it. Because of their special structure, these subset are called dyadic patterns. In this paper various properties of dyadic patterns are investigated. Exact upper bounds on approximation error are derived. In a simplest case, this error proves to be equal to $\frac{1}{6} log(n)$ for $n \times n$ sized images, as was conjectured previously by Goetz et al.