Natural representations for straight lines and the Hough transform on discrete arrays

The grid of discrete sampled data points in a digital image supports a limited set of lines at angles and displacements `natural´ to that grid. The effect of this implicit line quantization on the parametrization of the Hough transform is presented. The function describing the discrete Hough transforms line-detection sensitivity is derived. Expressions for the orientation, frequency, and popularity of lines in the natural set are given. The results obtained are of importance to data arrays of small size. The distribution of lines in the natural set is also important as it determines the precision and reliability with which straight lines can be measured on a discrete imaging array. From the natural line set concept, a general (a, d) slope/offset straight-line parametrization is developed for which the Hough transform is compact and fast to compute, and which is as easy to interpret as the class (p,θ) parametrization