Hypothesis testing: a framework for analyzing and optimizing Hough transform performance

In this paper a formal, quantitative approach to designing optimum Hough transform (HT) algorithms is proposed. This approach takes the view that a HT is a hypothesis testing method. Each sample in the HT array implements a test to determine whether a curve with the given parameters fits the edge point data. This view allows the performance of HT algorithms to be quantified. The power function, which gives the probability of rejection as a function of the underlying parametric distribution of data points, is shown to be the fundamentally important characteristic of HT behaviour. Attempting to make the power function narrow is a formal approach to optimizing HT performance. To illustrate how this framework is useful the particular problem of line detection is discussed in detail. It is shown that the hypothesis testing framework leads to a redefinition of the HT in which the values are a measure of the distribution of points around a curve rather than the number of points on a curve. This change dramatically improves the sensitivity of the method to small structures. The solution to many HT design problems can be posed within the framework, including optimal quantizations and optimum sampling of the parameter space. In this paper the authors consider the design of optimum I-D filters, which can be used to sharpen the peak structure in parameter space. Results on several real images illustrate the improvements obtained