In recent years, several methods of circle finding based on the Hough Transform (HT) have been proposed [1,2] as well as some general techniques for fast implementation of the HT [3,4]. Invariably these methods claim to improve efficiency, storage or reliability though in most cases the comparisons made with other techniques are superficial. We feel that this is about the right time to put a number of these algorithms together and examine their properties in more detail. The study is experimental and we consider both real and synthetic images. Our results show that more sophisticated variations of the HT method do not necessarily out-perform straightforward approaches.The paper is organised as follows. In the next section we introduce the circle finding problem and the basic idea underlying the HT. This is followed by a brief description of each of the five HT based methods considered in our study. The experimental evaluation of each method is then given and the final section presents the conclusions of our work.CIRCLE FINDING USING THE HT If a circle in the image is described as (1) where (a, 6) are the coordinate of the circle center and r is its radius, then an arbitrary edge point (i,,y,) will be transformed into a right circular cone in the (a, 6, r) parameter space [5]. If all the image points lie on a circle then the cones will intersect at a single point in (a, 6, r) corresponding to the parameters of the circle. Kimme et al [6] give probably the first known application of the Hough Transform to detecting circles in real images. In their work, they have made use of the direction of the gradient at each edge point. The centre of a circle must lie on the normal at the edge point. As a result instead of incrementing the whole circular cone, only segments of the cone need be incremented. The size the region which is incremented depends on the accuracy of the edge direction estimation.An important part of the complete HT process is peak detection. An extremely useful technique which we have found eases the peak finding problem considerably is the post-processing method proposed by Gerig and Klein [1]. It consists of a second daa pass which takes each edge point and identifies the maximum value in the accumulator array out of all parameter values voted for by the point. The edge point is labelled with this location. In all the methods considered, this technique is used to detect the final peaks. We refer the reader to [7,10] for details.
THE STANDARD HTThe Standard Hough Transform (SHT) in this study follows the basic idea outlined in the previous section. A 3-D accumulator array is employed and edge direction information is used to limit voting to a section of the cone. In an ideal situation, the centre of the circle must lie on a line oriented normal to the edge direction. Therefore we only have to move along the normal of every edge point to find the possible locations of centers. The distance between each edge point and the estimated center is a candidate for radius of the corresponding circle. However,...
The objective of this paper is to investigate a number of circle detection methods which are based on variations of the Hough Transform. The methods considered include the standard Hough Transform, the Fast Hough Transform of Li et al, two space saving approaches which are bused on those devised by Gerig and Klein and a twostage method. We experimentally compare the performance of the methods and illustrate properties such as accuracy, reliability, computational efficiency and storage requirements.
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