Deriving Stopping Rules for the Probabilistic Hough Transform by Sequential Analysis

It is known that Hough transform computation can be significantly accelerated by polling instead of voting. A small part of the data set is selected at random and used as input to the algorithm. The performance of these probabilistic Hough transforms depends on the poll size. Most probabilistic Hough algorithms use a fixed poll size, which is far from optimal since conservative design requires the fixed poll size to be much larger than necessary under average conditions. It has recently been experimentally demonstrated that adaptive termination of voting can lead to improved performance in terms of the error rate versus average poll size tradeoff. However, the lack of a solid theoretical foundation made general performance evaluation and optimal design of adaptive stopping rules nearly impossible. In this paper it is shown that the statistical theory of sequential hypotheses testing can provide a useful theoretical framework for the analysis and development of adaptive stopping rules for the probabilistic Hough transform. The algorithm is restated in statistical terms and two novel rules for adaptive termination of the polling are developed. The performance of the suggested stopping rules is verified using synthetic data as well as real images. It is shown that the extension suggested in this paper to A. Wald's one-sided alternative sequential test (Sequential Analysis, Wiley, New York, 1947) performs better than previously available adaptive (or fixed) stopping rules.