Number Theory Helps Line Detection in Digital Images

This paper deals with the problem of detecting line components in a digital image. For this purpose, the Hough Transform, which is based on voting in the dual plane, is widely used. However, there have been few theoretical studies on the relationship between its computational complexity and ability of detecting straight lines. In this paper we present two completely different algorithms for detecting every maximal line component contained in a digital image. The one, which is effective in the case of a dense digital image, is based on a new transformation named L1-Dual Transform defined by the L1-distance between points and lines. Number Theory supports efficient implementation of the algorithm. It can complete the required task in least time needed to achieve the above-mentioned ability of line detection. The other, which is effective when the edge density is low, attains efficiency by using the plane sweep technique in computational geometry. Furthermore, we present an efficient approximation algorithm which can detect at least α×100% of any maximal line component and show that its computational complexity depends on the value of α. Choosing α=0.5, for example, the time complexity of the algorithm is reduced from O(N4) to O(N3), where N is the length of one side of an image.