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.

翻译：快速霍夫变换是模式识别中广泛应用的算法。该算法依赖于一种称为二元线的特定离散线模型来近似直线。二元线与其所构造的理想直线之间的最坏情况偏差随图像线性尺寸n以O(log(n))增长，但实际达到最坏情况的直线极少。本文针对二元线与其理想对应线之间的偏差进行统计分析。具体而言，我们的研究发现平均偏差为零，方差随n以O(log(n))增长。随着n增大，这些（适当归一化的）偏差的分布趋近于均值为零、方差较小的正态分布。这一极限结果本质依赖遍历理论。