Computational comparison of voting-based and arrangement-based schema for digital line detection

The problem of detecting all possible digital line components contained in a given binary edge image is one of the most fundamental problems in pattern recognition and a great number of algorithms have been proposed under the name of Hough Transform [3, 4]. Most of them are based on the voting technique on a subdivided parameter space. Our basic idea is as follows: When we parameterize a line passing through an edge point by the angle and the distance to the line from the origin, the edge point is mapped to a sinusoidal curve in the ( ; ) parameter space. The intersection between two such curves corresponds to the line passing through the edge points. Thus, what we have to do is to compute intersections where a number of curves intersect. However, Since edge points are located on integer grids, it rarely happens that many edge points lie exactly on a line. We want to detect a set of edge points which lie roughly on a line. For this purpose we partition the parameter plane into small regions called buckets by axis-parallel lines, and for each such bucket we count the number of lines which pass through the bucket region. If the number of such lines exceeds a predetermined threshold, we report a digital line component corresponding to the bucket region. In computational geometry the dual transform between points and lines is more common than the above-described transform. It transforms a point (a; b) into a line y = ax + b and a line y = cx + d into a point (c; d). Again, an intersection between two lines for two edge points corresponds to a line which passes through these two edge points. Thus, if we enumerate all intersections at which many lines intersect, we can detect all digital line components. This voting technique is easy to be programmed and also easy to be tuned for practical use. This is why the Hough transform is commonly used. However, there are many problems to be resolved. For example, an optimal way of partitioning the parameter plane is somewhat strange and not good for e cient computation. In this paper we analyze the voting technique based on the Hough transform from a standpoint of computational complexity following a mathematical de nition of a digital line component to be detected. We also present a new algorithm for detecting line segments with two endpoints considering their point density.